The Ehrhart polynomial ehrP(n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f∗-vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of ehrP(n) with respect to the binomial coefficient basis (Formula presented.), where d = dimP. Similarly to h/h∗-vectors, the f∗-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f∗-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f∗-coefficients increases and the last quarter decreases. Even though f∗-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h∗-vector, there is a polytope with the same h∗-vector whose f∗-vector is unimodal.