​In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most effective mixed differences to include in the MISC estimator. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi-Index and Multi-Level Monte Carlo, Multi-Level Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods.