Abstract: Like the ring of algebraic integers, the ring Int(Z) of polynomials in Q[X] assuming integer values at all of the integers is an example of a non-Noetherian integrally closed domain of finite Krull dimension. Like Z[X], the domain Int(Z) has Krull dimension 2. Nevertheless Int(Z) is still very much like a Dedekind domain. For example, the nonzero finitely generated fractional ideals of Int(Z) form a group under multiplication, and the finitely generated ideals of Int(Z) are two-generated. In this talk I will present some new results and open problems about more general "integer-valued polynomial rings." I will also introduce some classes of "Dedekind-like" integrally closed domains, such as the Prufer domains, Krull domains, and Prufer v-multiplication domains.