The partially disjoint paths problem asks for paths $P_1, \ldots,P_k$ between given pairs of terminals, while certain pairs of paths $P_i$,$P_j$ are required to be disjoint. With the help of combinatorial group theory, we show that, for fixed $k$, this problem can be solved in polynomial time for planar directed graphs. We also discuss related problems. No specific foreknowledge is required.