We say that the sequence $(a_n)$ is quasi-polynomial in $n$ if there exist polynomials $P_0,...,P_{s-1}$ and an integer $n_0$ such that, for all $n\ge n_0$, $a_n=P_i(n)$ where $i\equiv n\pmod{s}$. We present several families of combinatorial objects with the following properties: Each family of objects depends on two or more parameters, and the number of isomorphism types of objects is quasi-polynomial in one of the parameters whenever the values of the remaining parameters are fixed to arbitrary constants. For each family we are able to translate the problem of counting isomorphism types of objects into the problem of counting integer points in a union of parametrized rational polytopes. The families of objects to which this approach is applicable include combinatorial designs, linear and unrestricted codes, and dissections of regular polygons.