A basic question in geometry and topology is to discover necessary (and perhaps sufficient) conditions for two manifolds to be locally or globally equivalent for some notion of equivalence. An example of a global topological invariant is the fundamental group of a topological space, because having isomorphic fundamental groups is a necessary condition for two spaces to be equivalent up to homotopy. We restrict ourselves to real hypersurfaces in $C^2$. I'll sketch Poincare's proof of the global inequivalence of the unit ball and polydisc, then outline a method due to Cartan, Chern, and Moser, about how to find a system of invariants for the local equivalence problem. Knowing a bit of differential geometry and complex analysis would be helpful, but isn't essential.