I will discuss how to study commuting sets of operators on Hilbert space which satisfy polynomial relations. Under a natural norm constraint, there is a universal operator algebra that models this. In an effort to classify these algebras up to isomorphism, one must deal with the variety associated to the polynomial relations. Classification up to completely isometric isomorphism is very nice. But the algebraic isomorphism problem raises many difficulties. Ideas from algebraic geometry are combined with a function theoretic representation of our algebras as multipliers on a Hilbert space of functions on the variety. (This is joint work with Orr Shalit and Chris Ramsey.)