Giordano Bruno (1548-1600) has gone down in history more as a legendary figure and as a “martyr for science”, than for making any substantial contributions to scientific thought. He read and wrote extensively, producing numerous papers and books, including works on mnemonics, philosophy and astronomy, though the consensus is that he was not very informed in the latter and had a tendency towards too much speculation. He was among the few philosophers of his day to accept the Copernican heliocentric model, but also speculated about an infinite universe with numerous inhabited worlds and believed in metempsychosis and transmigration, basically reincarnation.  He spent a large portion of his life as a wanderer from country to country all over Europe avoiding an Inquisitional indictment against him for blasphemy and heresy. At the age of 17 he entered the Dominican Order, and was ordained a priest at 24, but only four years later began his wanderings after fleeing from his Inquisitors.
So what is so interesting about Bruno? Basically his taste for freethinking and the forbidden, and his eventual burning at the stake in 1600 by the Inquisition. Had this not happened it’s debatable whether Bruno would have drawn so much attention. As one commentator noted:
Bruno was born five years after Copernicus died. He had bequeathed an intoxicating idea to the generation that was to follow him. We hear a lot in our own day about the expanding universe. We have learned to accept it as something big. The thought of the Infinity of the Universe was one of the great stimulating ideas of the Renaissance. It was no longer a 15th Century God's backyard. And it suddenly became too vast to be ruled over by a 15th Century God. Bruno tried to imagine a god whose majesty should dignify the majesty of the stars. He devised no new metaphysical quibble nor sectarian schism. He was not playing politics. He was fond of feeling deep thrills over high visions and he liked to talk about his experiences. And all of this refinement went through the refiners' fire -- that the world might be made safe from the despotism of the ecclesiastic 16th Century Savage. He suffered a cruel death and achieved a unique martyr's fame. He has become the Church's most difficult alibi. She can explain away the case of Galileo with suave condescension. Bruno sticks in her throat.
He is one martyr whose name should lead all the rest. He was not a mere religious sectarian who was caught up in the psychology of some mob hysteria. He was a sensitive, imaginative poet, fired with the enthusiasm of a larger vision of a larger universe ... and he fell into the error of heretical belief. For this poets vision he was kept in a dark dungeon for eight years and then taken out to a blazing market place and roasted to death by fire.
It is an incredible story.
The "Church" will never outlive him.
Thus we can see why Bruno fires the popular imagination, though even Galileo didn’t think very highly of him. Some may well say that Bruno engaged in “magical thinking”, and he was influenced by Arab astrological magic, Neoplatonism and Hermeticism, basically all speculative religious philosophies.
One of the more interesting criticisms of Bruno comes from Prof. Richard W. Pogge in The Folly of Giordano Bruno :
In popular accounts of the life of Bruno, it is often said that he was condemned for his Copernicanism and his belief in life on other worlds. He is portrayed as a martyr to free thought, and an early, prosecuted proponent of the modern view of the universe, hounded across Europe by the Inquisition for his beliefs and finally paying the ultimate price for them in a fiery public death. He has become a symbol of the intolerance of authority in the face of new ideas. These accounts, however, often leave out two fundamental aspects of the case of Giordano Bruno that cast matters in a somewhat different light. The first calls into doubt how closely we should link Bruno with the history of astronomy and what came to be called the "Scientific Revolution", and the second offers a perspective on the undeniable tragedy of his life that make him less of a symbol, but in the balance makes him more human.
The one key fact of the study of Bruno's life is that we do not actually know the exact grounds of his conviction on charges of heresy. The simple reason is that the relevant records have been lost. This is quite unlike the state of affairs in the later trial of Galileo, where we have extensive documentation including the forgeries that played a role in the case against him. In the case of Bruno, we must seek clues in contemporary accounts and in an examination of his writings.
Pogge also argued that Bruno was a victim of his own outspoken and contentious personality:
The second often overlooked fact of Bruno's life concerns his period of exile between 1576 and 1591. Most brief popular accounts state the bare facts of his peregrinations around Europe, but what is left unsaid is that his wanderings appear to have had less to do with his being hounded by the Inquisition as it did with his own rather difficult personality. While Bruno was fairly successful for a time at finding powerful and sympathetic patrons to shelter him, he invariably did something to alienate and outrage them, usually fairly quickly after entering their service. The Inquisition had little to do with it, as once he left Italy, he was effectively out of their reach. This was especially true of his time spent under the protection of the French Ambassador to protestant England (1583-85) during the reign of Queen Elizabeth I, and his wandering around protestant Germany.
An examination of his actions during this period of exile makes clear that almost all of his misfortunes were brought down upon himself without the Inquisition's help. He outraged the faculty at Oxford with his lectures, he became embroiled in violent quarrels over trivial matters, and generally succeeded in alienating those people best able to protect him. His actions during this period reveal the very hallmark of folly, namely repeated failure to act in his own best interests even when reasonable alternatives were available. His final return to Italy (which resulted in his arrest in Venice a year later) can be seen as being motivated in part by the fact that by 1591 he had effectively burned most of his bridges behind him and thus he had little choice. In many ways, Bruno thrust himself into the flames that rose into the winter skies of the Campo di Fiore on the 17th day of February in 1600.
Bruno was brilliant, contentious, and ultimately self-destructive. There is nothing in his writings that contributed to our knowledge of astronomy in any substantial way, indeed his astronomical writings reveal a poor grasp of the subject on several important points. I think we pay attention to him today in large measure because among other things he vocally espoused (but apparently did not really understand) Copernicanism, an idea which was to become the key insight that led to our view of the world. In addition, his On the Infinite Universe and Worlds appeals to many today because of its apparent resonance with the deeply held conviction that life exists elsewhere in the Universe, despite the fact that proponents of extraterrestrial life would find little of interest within its difficult pages. It also does not hurt his mystique that he came to a rather spectacular and violent ending, ostensibly in punishment for these beliefs by the reigning authorities of his day. In the end, Bruno bet on the right horse (if perhaps for questionable reasons), and thus has become a kind of culture hero instead of a footnote in books on Renaissance philosophy.
A more hagiographical view of Bruno comes from Frank Gaglioti :
The current attitude of the Roman Catholic Church to Bruno is defined by a two-page entry in the latest edition of the Catholic Encyclopaedia. It describes Bruno's "intolerance" and berates him, declaring "his attitude of mind towards religious truth was that of a rationalist”. The article describes in detail Bruno's theological errors and his lengthy detention at the hands of the Inquisition, but fails to mention the best-known fact—that the church authorities burnt him alive at the stake.
Bruno has long been revered as a martyr to scientific truth. In 1889 a monument to him was erected at the location of his execution. Such was the feeling for Bruno that scientists and poets paid tribute to him and a book was written detailing his life's work. In a dedication for a meeting held at the Contemporary Club in Philadelphia in 1890, American poet Walt Whitman wrote: "As America's mental courage (the thought comes to me today) is so indebted, above all current lands and peoples, to the noble army of old-world martyrs past, how incumbent on us that we clear those martyrs' lives and names, and hold them up for reverent admiration as well as beacons. And typical of this, and standing for it and all perhaps, Giordano Bruno may well be put, today and to come, in our New World's thankfulest heart and memory."
According to the Catholic Encyclopedia
Bruno was not condemned for his defence of the Copernican system of astronomy, nor for his doctrine of the plurality of inhabited worlds, but for his theological errors, among which were the following: that Christ was not God but merely an unusually skilful magician, that the Holy Ghost is the soul of the world, that the Devil will be saved, etc……..And one can readily understand how Bruno's effort to establish a unitary concept of nature commanded the admiration of such men as Spinoza, Jacobi, and Hegel. On the other hand, the exaggerations, the limitations, and the positive errors of his scientific system; his intolerance of even those who were working for the reforms to which he was devoted; the false analogies, fantastic allegories, and sophistical reasonings into which his emotional fervour often betrayed him have justified, in the eyes of many, Bayle's characterization of him as "the knight-errant of philosophy." His attitude of mind towards religious truth was that of a rationalist. Personally, he failed to feel any of the vital significance of Christianity as a religious system. It was not a Roman Inquisitor, but a Protestant divine, who said of him that he was "a man of great capacity, with infinite knowledge, but not a trace of religion."
Nevertheless the case against Bruno by the Inquisition as outlined by Luigi Firpo  is as follows:
• Holding opinions contrary to the Catholic Faith and speaking against it and its ministers.
• Holding erroneous opinions about the Trinity, about Christ's divinity and Incarnation.
• Holding erroneous opinions about Christ.
• Holding erroneous opinions about Transubstantiation and Mass.
• Claiming the existence of a plurality of worlds and their eternity.
• Believing in metempsychosis and in the transmigration of the human soul into brutes.
• Dealing in magics and divination.
• Denying the Virginity of Mary.
According to the Catholic Encyclopedia Bruno offered a full recantation in 1591, but was then extradited from Venice to Rome for further examination by the Inquisition. Apparently over the next eight years of imprisonment Bruno still stood by his views. The Roman Catholic inquisitor Cardinal Bellarmine is supposed to have again demanded a full recantation, which Bruno refused. At the conclusion of his trial, with threatening gestures towards his judges, Bruno said, “"Perchance you who pronounce my sentence are in greater fear than I who receive it."
He was declared a heretic and handed over to secular authorities, and on 17 February 1600 he was burned at the stake, his ashes dumped in the river Tiber, and all of his works placed in the Index Librorum Prohibitorum in 1603.
Interest in Bruno was not revived until the 18th century, and in the 19th century Italian scholars began to be intrigued by him. In 1889 a monument was erected to him at the site of his execution. His popularity has since become worldwide, and even here in Australia the popular radio station 2GB was named after Giordano Bruno when it was started by the Theosophical Broadcasting Station Pty Ltd. , which was owned by a local branch of the Theosophical Society Adyar. Even on Facebook Bruno has nearly 18,000 fans. While Richard Dawkins has over 42,000 fans, cosmologist Paul Davies and philosopher Daniel Dennett (278 and 1,031 respectively) come no where near Bruno’s popularity, yet there is no doubt that both Davies and Dennett espouse much more realistic and scientific views. So once again, we see that in the popular imagination Bruno stands for something close to the heart of ordinary people, perhaps breaking taboos, questioning authority, shunning religious control and authoritarianism, thinking new and unorthodox ideas, even if not rubber-stamped by contemporary science. Perhaps in some distant future more of Bruno’s ideas will find justification. Is reincarnation a fact? Are there numerous inhabited worlds, and is the universe infinite and ever-expanding? Do we have eternal souls that can never die? These questions, once braved by Bruno, still lie outside the scope of science. Thankfully, the days of heretic-burning are long gone. Too late for Giordano Bruno, but his questing and inquisitive spirit lives on in the hearts of many. Unorthodox, offensive, contentious and stubborn may be apt descriptions of Bruno, but none of this has dimmed his popularity, maybe even enhanced it.
This whole which is visible in different ways in bodies, as far as formation, constitution, appearance, colors and other properties and common qualities, is none other than the diverse face of the same substance — a changeable, mobile face, subject to decay, of an immobile, permanent and eternal being. – Giordano Bruno
Transmigration: “In transmigration after death, the soul, or shade, drinks from the river Lethe and loses all past memories of their previous life while in Hades, or underworld, and then moves (or transmigrates) into another human form and is reborn. It was thought the soul had been, and always would be, eternal, having no beginning or end.” (source: Wikipedia)
 Luigi Firpo, Il processo di Giordano Bruno, 1993.