Non-commutative probability is an analogue of probability theory invented by Dan Voiculescu in which the random variables do not commute but there is a suitable notion of “integration.” I will motivate the study of this modern subject and explain applications of my work and others to the study of random matrices and C* and von Neumann algebras (the natural replacement for functions on a probability space). No C* algebra or von Neumann algebra background will be assumed in this talk.