The theory of integer partitions is a rich subject that lives in the intersection of number theory and combinatorics.  In this colloquium-style talk, I will go through a brief history of partitions and the various tools used to study them, along with connections to Waring's problem and other topics in additive number theory.  I will then state some results about counting partitions in which the parts are restricted to various subsets of the integers (e.g., primes, squares, arithmetic progressions).