We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs
that exploit the particular topology of the input graph. An important feature of our algorithms is that they can
work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. In the
case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time.
A distance query is also answered in logarithmic time. In the case of planar digraphs, we give an interesting
tradeoff between preprocessing, query, and update times depending on the value of a certain topological
parameter of the graph. Our results can be extended to n-vertex digraphs of genus O.n1¡"/ for any " > 0.