An introduction to computability theory (also known as recursion theory). A discussion of the problem of determining what it means for a set or function to be computable, including primitive recursion, Turing machines, and the Church-Turing Thesis. The theory of Turing degrees and the computably enumerable sets. Topics: the halting set, Turing reducibility and other reducibilities, Post's problem, the Recursion Theorem, priority arguments, and more.