The classical theory of total positivity concerns matrices in which all minors are nonnegative. While this theory was pioneered in the 1930's, the past decade has seen a flurry of research in this area initiated by Lusztig, who extended the theory to flag varieties and real reductive groups. In this talk we will describe the combinatorics and geometry of the totally nonnegative part of the real Grassmannian, and of its cell decomposition. Its cell decomposition has many interesting descriptions, in terms of certain tableaux, in terms of certain permutations, etc, and the tableaux description allows us to give generating functions enumerating cells according to dimension. One corollary of this work is a new q-analog of the Eulerian numbers, which interpolates between the binomial coefficients, the Narayana numbers, and the Eulerian numbers.