Chung Ling Soo was a stage character created by a Caucasian American man, William Ellsworth Robinson, who disguised himself as a Chinese man to cash in on audiences' enthusiasm for the exotic. Robinson lived as Chung, never breaking character while in public. He died in March 1918 when a bullet catch trick went wrong. "My God, I've been shot" were both his last words and the first English he had spoken on stage in 19 years.
Nikola Tesla was a world-renowned inventor, physicist, and engineer. For a while, he conducted electrical experiments at his lab in Colorado Springs, where he was also known for his eccentric behavior.
When Andy Serkis' character is first introduced, he makes a reference to a magic trick where you guess the item in a person's pocket. This is the exact same trick that fooled Serkis' character Gollum in The Hobbit: An Unexpected Journey (2012).
In the Bullet Catch scene, you can clearly see the name Harry Dresden on the list of performers under "The Professor." Harry Dresden is a fictional wizard in "The Dresden Files," a series of books by novelist Jim Butcher, and later the basis of The Dresden Files (2007).
This is one of two Christopher Nolan's movies in which a character has a two-head coin, the other one being The Dark Knight, in which Harvey Dent/Two-Face has one, used to decide his fate or the fate of his victims.
When Nikola Tesla's machines are being exhibited in the Royal Albert Hall, a man in the audience protests that Tesla's electrical current is unstable. Later in the movie, the same man appears in Colorado Springs as one of Thomas A. Edison's henchmen, thus proving that magicians are not the only ones who hide within their rivals' audiences.
Root, the on-stage double of Angier (Hugh Jackman), announces that he has played Faust and Caesar in the past. Both were famously portrayed on stage as men destroyed by their own ambition, as Angier eventually is.
Angier's double mumbles a few lines from a speech while rehearsing on stage before his first performance. What he's saying is actually the words of Harry Percy (Hotspur) from William Shakespeare's Henry IV, when called to appear before the king and explain his failure to turn over prisoners after a recent battle in Scotland. Apparently Hugh Jackman has used this speech in previous auditions. Presumably it was believed that having the double deliver a few lines from Shakespeare would lend him an actorly air, as his character is in fact a dissolute stage actor.
There is a clue to the fact that Christian Bale's character Borden has a double early on in the movie, when he says, "I would not forgive 'myself' for selling my own trick" when referring to Fallon's decision not to sell the Transported Man secret to Owens. He should have said, "I would not forgive 'him' for selling my own trick," but as Fallon is a double of Borden, then he is referring to him as himself.