Briefly, the background of the Bavarian Illuminati puzzle is this. On May 1, 1776, in Bavaria, Dr. Adam Weishaupt, a professor of Canon Law at Ingolstadt University and a former Jesuit, formed a secret society called the Order of the Illuminati within the existing Masonic lodges of Germany. Since Masonry is itself a secret society, the Illuminati was a secret society within a secret society, a mystery inside a mystery, so to say. In 1785 the Illuminati were suppressed by the Bavarian government for allegedly plotting to overthrow all the kings in Europe and the Pope to boot. This much is generally agreed upon by all historians. Everything else is a matter of heated, and sometimes fetid, controversy.
For almost every secret society there exists some theorised link to the Illuminati, however spurious. Whether the Illuminati was just a small secret society that inspired many others or whether it continues to this day behind the scenes is a matter of much debate. Conspiracy theorists are often quick to point out that the Eye-in-the-Pyramid design on the Great Seal of the United States (also printed on the back of the U.S. Dollar note) is a Masonic symbol.