This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given a finite set E and k weighted matroids (E, mathcal{I}_i, w_i), i = 1, dots, k, and our task is to find a minimum partition (I_1,dots,I_k) of E such that I_i in mathcal{I}_i for all i. For each objective function, we give a polynomial-time algorithm or prove NP-hardness. In particular, for the case when the given weighted matroids are identical and the objective function is the sum of the maximum weight in each set (i.e., sum_{i=1}^kmax_{ein I_i}w_i(e)), we show that the problem is strongly NP-hard but admits a PTAS
