This talk will give an overview of the Zariski-Lipman theory of complete ideals in the local ring of a point on a smooth surface. In particular, we will talk about proximity matrices and their use in computations of embedded resolutions in two dimensions. We will also talk about the relation to the germs of analytically irreducible curves. If there is time, we will mention Jarvilehto's use of the theory to compute the jumping numbers of these ideals.