Since you tag this question with "Ancient Rome" and "Galileo," you must be referring to the fictitious version of the Illuminati. There was a real group that was called the Illuminati, but this was an Enlightenment era group that was founded in 1776 and could have had nothing to do with Galileo, much less ancient Rome.
The fictitious Illuminati are a secretive group that has existed (to some people such as the writer of this link http://www.illuminati-news.com/moriah.htm) since the start of human history. The Illuminati are said to be controlling world events pushing to create a single world government under their control. They are behind all great world events. There is, however, no real evidence for the existence of such a group.
They are a group of families on Earth who have more divine blood in them than the rest of humanity and rule us behind the scenes. Back when the Gods were on the Earth, Kingship was granted to men who were descended from the Gods (called Demigods). These people always interbred with one another to preserve their divine blood and claimed to be elligable to rule due to their "Divine Right". With the absence of the Gods today, the Illuminati are the ones chosen and tasked to push on the agenda until the Gods return. The Illuminati are more wealthy than anyone can imagine and have more power and influence than world leaders (these world leaders such as our President are put into power by the Illuminati to be used as spokesmen). They have existed essentially since the Gods granted Kingship to humanity. More recently in history, the Knights Templar, who were amassing wealth and power in France, came to Jerusalem under Solomon's rule to secretly look for "The Holy Grail", as well as to search for an unspeakable amount of wealth supposedly located under temple mount. According to Illuminati theorists, the Holy Grail was the offspring of Jesus Christ. The Knights Templar were said to have found this Holy Grail (as well as this wealth). If this Grail was truly the bloodline of Christ, then the Illuminati wanted to capture it in order to once again re-purify themselves in a Divine Bloodline to assert their Divine Right over the populace.
America was founded by the Illuminati (Freemasons, which are an outlet of the Illuminati) and is a testing ground for the Single World Government which they are pushing for. Pearl Harbor was allowed to happen so that we could enter WWII, a war controlled by the Illuminati on both sides of the conflict. After WWII, our exhausted nations decided to form the United Nations (which is a step towards the Single Word Government). 9/11 was planned and executed by the Illuminati to take away our freedoms and to start the WWIII which will be a war aimed to depopulate the Earth by nearly 95%, so that the Illuminati may assert their dominance over the easily-controlled few. The Bible's "Anti-Christ" will come after the dropping of the Nuclear Weapons and will bring us all into a single world government. The reason they want this government is because back many thousands of years ago, Gods ruled separate sections of the Earth. Enki/Poseidon ruled Africa, Enlil/Zeus ruled Mesopotami, Innana/Ishtar ruled India, ect... The Illuminati were told by a God who wanted to claim supremacy over Earth (the New Testament is a campaign for this God) to form the nations into a Single Government so that a Single God may rule it.
The Illuminati members can be found if you look closely at who is really pulling the strings. Look for people who are sickenly wealthy, have ties to politics, are behind multiple corporations, or news/media outlets. According to theorists, the names Bush, Disney, McDonald, Rockefeller, Kennedy/Cavendish, Hamilton, Sinclair, Rothschild/Bauer, and Windsor have ties to the Illuminati.
If you refuse to accept this terrifying truth about the Illuminati, or just would rather not think about it, I urge you to at least accept the fact that those who appear to be in power (presidents, dictators, congressmen, ect) are not the actual ones with the power, they are just faces. The real ones who control this Earth are the ones with the money...