An algebraic number is a root of a polynomial with integer coefficients. There is a natural generalization of what it means to be an integer for such numbers, and we address the question: what are the smallest such algebraic integers, excluding zero? There are several different ways to make this question precise: one formulation is easy, another formulation has no answer at all, and a third formulation leads to an open problem which has been unresolved for 80 years. We discuss some simple variants, and describe some recent progress.