We consider the problem of approximating low eigenvalues of the Laplace operator on bounded domains in n dimensional Euclidean space with Dirichlet boundary conditions. The general purpose is to be able to understand better the relationships between the geometry of the domain and low eigenvalues, and we divide our approach into (roughly) three parts as follows: 1) asymptotic expansions 2) bounds depending on geometric quantities 3) more complex conjectured bounds supported by extensive numerical computations