Let f be a polynomial with integer coefficients. The monodromy conjecture predicts a relationship between the Igusa zeta function of the hypersurface V(f), which governs the number of solutions to f = 0 (mod p^n) for a prime p, and the eigenvalues of the monodromy action on the cohomology of the Milnor fiber, which is a topological invariant of the complex hypersurface. When f is nondegenerate with respect to its Newton polyhedron, which is true for "generic" polynomials, there are combinatorial formulas for both the Igusa zeta function and the eigenvalue of monodromy. I will describe recent results (joint with S. Payne and A. Stapledon) which prove a version of the monodromy conjecture for nondegenerate polynomials which have a simplicial Newton polyhedron.