Illuminati conspiracy: identity, theories, and religion
The original Illuminati was an 18th century secret society in Europe that sought to drastically, yet subversively, affect society. In recent history, some theorists have offered evidence suggesting that the Illuminati never really disbanded like they were ordered to do (more below), but have continued to exist mostly undetected in the world.  Skeptics say those ideas are just unfounded conspiracy theories.
What is the Illuminati accused of doing in the world today?
Accusations of the group's activities today range from orchestrating the takeover of the world's financial systems to controlling politicians, even entire governments. They are also charged with developing and advancing various schemes through massive corporations to achieve different objectives, which often involve taking over large segments of society. Some also suspect them of behaviors associated them with the occult.
What ties does the Illuminati have to religion?
It has several. First, the original Illuminati sought to suppress religion's role in society. Second, and as a result of the first tie, the secret society had a contentious relationship with the Roman Catholic Church, which was influential in the society's demise in the 18th century.  Also, for believers of present-day Illuminati theories, the society is either in opposition to the Catholic church (and other organized religions) or it is intertwined with the Church in a secretive way. Third, some have connected the Illuminati to modern expressions of the occult and Satanism.
History of the Illuminati
What was the original Illuminati like?
Originally called the "Bavarian Illuminati" ("Bavarian" refers to a region of Germany), this secret society was established on May 1, 1776 by Adam Weishaupt (1748-1830). Weishaupt was a philosopher, following the teachings of John Locke and the school of Empiricism (i.e. adherents believe that all knowledge is based on the senses alone). He was known as "Brother Spartacus" within the order. Weishaupt was associated with freemasonry and many have noted the similarities between the orders. 
Why did the original Illuminati exist?
One of the objectives of the Illuminati was to reduce the impact of suppressive regimes (e.g. religious or political leaders and systems). In particular, the society sought to reduce the influence of the Roman Catholic church.  The Illuminati was also at odds with some of the traditional philosophical underpinnings of the Christian faith.
What well-known people were a part of the Illuminati?
Duke Ferdinand, Prince of Brunswick-Lüneburg (1721-1792, known for his participation in the Seven Years' War); Franz Xaver von Zwackh (1756-1843, known for being the first President of the Government of the Bavarian Rhine circle in Speyer); Johann Wolfgang von Goethe (1749-1832, known for being a writer and artist); and Johann Gottfried von Herder (1744-1803, known as a philosopher, theologian, poet, and literary critic).
What led to the group's public downfall?
In 1777, Karl Theodor, ruler of Bavaria, banned secret societies like the Illuminati. Weishaupt is reported to have left the group and the area in 1785. In 1787, von Zwack's home was searched and writings of the society were found, which exposed the group's ambitions.
How big was the original group?
The Illuminati has groups in many European countries. At its public peak, the society reportedly had around 2,000 members.
Modern Illuminati theories
What about modern "Illuminati" groups?
Some groups today use the name Illuminati, and a few claim to be the rightful descendents of the 18th century group. Most outsiders question those claims. Other groups like the Ordo Templi Orientis, use related terminology, even the word "illuminati," but their labels have different meanings than the original society. Some of these groups seek privacy like the original order did, but others make efforts to acquire attention.
What authors have written about potential modern-day Illuminati?
Mark Dice may be the most well-known. His YouTube channel has 55 million views. David Icke, Texe Marrs, Jüri Lina and Morgan Gricar have argued that the Bavarian Illuminati have survived, possibly to this day.
What do theorists accuse the Illuminati of doing?
Accusations are wide-ranging. Some allege that the Illuminati has gotten various U.S. presidents elected. Others see the secret society responsible for different events like the September 11, 2001, terrorist attacks on the U.S. and the assassination of president John F. Kennedy and the French Revolution. Others believe they control the entertainment industry in Hollywood.
Is Jay Z a member of the Illuminati?
In 2013, internet rumors circulated that rapper Jay Z was a member of the Illuminati. These accusations were mostly based on symbols that have been linked to the Illuminati, which were found in some of his videos.
McKeown, Trevor W. "A Bavarian Illuminati Primer". Grand Lodge of British Columbia and Yukon A.F. & A.M.
Catholic encyclopedia: "Illuminati" (http://www.newadvent.org/cathen/07661b.htm )
Goeringer, Conrad. "The Enlightenment, Freemasonry, and The Illuminati". American Atheists.