Almost every problem on digraphs requires computing strongly connected components and directed spanning trees in one form or another. It has long been an open problem whether polylog time and linear processors are enough to find the strongly connected components of a digraph and compute directed spanning trees for these components. This paper provides the first non-trivial partial solution to this open problem: For a planar digraph with n vertices, the strongly connected components can be computed in O(log3 n) time and O(n) processors. If the graph is strongly connected, a directed spanning tree can be built in O(log2 n) time and O(n) processors. Both algorithms are deterministic and run on a parallel random access machine that allows concurrent reads and concurrent writes in its shared memory.