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Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property (Pauli's exclusion principle) that the number of points in any subregion is a sum of independent Bernoulli random variables. Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We use these facts to give simple probabilistic proofs of existence criteria for determinantal processes and central limit theorems. Analogous results will be given for permanental processes with geometric variables replacing the Bernoulli variables. This is joint work with Manjunath Krishnapur, Yuval Peres and Balint Virag.