Classical black holes are defined by the property that things can go in, but don't come out. However, Stephen Hawking calculated that black holes actually radiate quantum mechanical particles. The two important ingredients that result in back hole evaporation are (1) the spacetime geometry, in particular the black hole horizon, and (2) the fact that the notion of a "particle" is not an invariant concept in quantum field theory. These notes contain a step-by-step presentation of Hawking's calculation. We review portions of quantum field theory in curved spacetime and basic results about static black hole geometries, so that the discussion is self-contained. Calculations are presented for quantum particle production for an accelerated observer in flat spacetime, a black hole which forms from gravitational collapse, an eternal Schwarzschild black hole, and charged black holes in asymptotically deSitter spacetimes. The presentation highlights the similarities in all these calculations. Hawking radiation from black holes also points to a profound connection between black hole dynamics and classical thermodynamics. A theory of quantum gravity must predicting and explain black hole thermodynamics. We briefly discuss these issues and point out a connection between black hole evaportaion and the positive mass theorems in general relativity.