The theory of determinantal point processes has its roots in work in mathematical physics in the 1960s, but it is only in recent years that it has been developed beyond several specific examples. While there is a rich probabilistic theory, there are still many open questions in this area, and its applications to statistics and machine learning are still largely unexplored. Our contributions are threefold. First, we develop the theory of determinantal point processes on a finite set. While there is a small body of literature on this topic, we offer a new perspective that allows us to unify and extend previous results. Second, we investigate several new kernels. We describe these processes explicitly, and investigate the new discrete distribution which arises from our computations. Finally, we show how the parameters of a determinantal point process over a finite ground set with a symmetric kernel may be computed if infinite samples are available. This algorithm is a vital step towards the use of determinantal point processes as a general statistical model.