Given an undirected graph G=(V,E) with nonnegative weights on the vertices, the feedback vertex set problem is that of finding a minimum-weight set T of vertices such that every cycle of the graph contains a vertex in T. Several variants of this problem have been studied, including those in which one must find a minimum-weight set of vertices T such that: (1) every directed cycle of a directed graph contains a vertex in T (the directed feedback vertex set problem); (2) every odd length cycle contains a vertex of T (the graph bipartization problem); (3) every cycle containing a vertex of a given set S contains a vertex of T (the subset feedback vertex set problem). All of these problems are NP-hard, even in planar graphs. In this talk, we show how the primal-dual method for approximation algorithms can be used to find a solution of value within 9/4 times the value of an optimal solution for each of these problems in the case of planar graphs. Joint work with Michel Goemans.