Counting Integer Points in Higher-Dimensional Polytopes

We survey some computationally efficient formulas to estimate the number of integer or 0-1 points in polytopes. In many interesting cases, the formulas are asymptotically exact when the dimension of the polytopes grows. The polytopes are defined as the intersection of the non-negative orthant or the unit cube with an affine subspace, while the main ingredient of the formulas comes from solving a convex optimization problem on the polytope.