The Feynman path integral interprets quantum amplitudes as sums over all possible histories which are "weighted'' by the classical action. In interpreting that statement, I'll review some classical and quantum mechanics, and then convince you why the Feynman path integral is reasonable. I'll then convince you that everything I just said is mathematical nonsense. I'll also try to indicate some nice math that has come out of attempting to make sense of this over the past fifty years. I'll try to make the lecture as self-contained as possible, but familiarity with basic concepts of classical and quantum mechanics would be helpful. \textbf{Warning}: This talk contains lots of hand waving.