We develop an algorithmic theory of nonlinear optimization over sets of integer points presented by inequalities or by oracles. Using a combination of geometric and algebraic methods, involving zonotopes, Graver bases, multivariate polynomials and Frobenius numbers, we provide polynomial-time algorithms for broad classes of nonlinear combinatorial optimization problems and integer programs in variable dimension. I will overview this work, joint with many colleagues over the last few years, and discuss some of its many applications in statistics and operations research, including privacy in statistical databases, experimental design, nonlinear transportation, multicommodity flows, error-correcting codes, matroids and general independence systems.