I will describe a multivariate generalization of the classical hypergeometric equations, based on the concept of a toric ideal. A central question about these A-hypergeometric systems concerns the behavior of the solutions as the parameters vary. Although much progress has been made on special cases, the general solution seemed far away. A new conjecture, joint with Ezra Miller, involving the local cohomology of the underlying toric ring, has shed light on this question. T