A notion of curvature is introduced in multivariable operator theory. The curvature invariant of a Hilbert module over C[z(1),., z(d)] is a nonnegative real number which has significant extremal properties, which tends to be an integer, and which is hard to compute directly. It is shown that for graded Hilbert modules, the curvature agrees with the Euler characteristic of a certain finitely gen...