Equivariant cohomology theories are robust invariants for studying spaces with an action by a group G. In contrast to classical cohomology theories, these invariants do not take values in abelian groups. Instead, there are several possible algebraic structures, known collectively as incomplete Mackey functors, which provide a home for these additive theories. Similarly, multiplicative equivariant cohomology theories can take values in one of several different generalizations of rings. In this talk I will discuss examples of these different algebraic structures, and some recent work on the interplay between the various multiplicative and additive structures that arise in equivariant topology.