Kernel functions have become an extremely popular tool in machine learning, with many applications and an attractive theory. This theory views a kernel as performing an implicit mapping of data points into a possibly very high dimensional space, and describes a kernel function as being good for a given learning problem if data is separable by a large margin in that implicit space. In this talk I will describe an alternative, more general, theory of learning with similarity functions (i.e., sufficient conditions for a similarity function to allow one to learn well) that does not require reference to implicit spaces, and does not require the function to be positive semi-definite (or even symmetric). In particular, I will describe a notion of a good similarity function for a given learning problem that (a) is fairly natural and intuitive (it does not require an implicit space and allows for functions that are not positive semi-definite), (b) is a sufficient condition for learning well, and (c) strictly generalizes the notion of a large-margin kernel function in that any such kernel is also a good similarity function, though not necessarily vice-versa.