We will discuss a generalization of Stanley's celebrated theorem that the h*-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h*-polynomial as a real-valued function for a larger family of weights. This is joint work with Esme Bajo, Robert Davis, Jesús A. De Loera, Alexey Garber, Katharina Jochemko and Josephine Yu.