In this paper we investigate the problem of adding a minimum number of edges to a planar graph in such a way that the resulting graph is biconnected and still planar. It is shown that this problem is NP-complete. We present an approximation algorithm for this planar biconnectivity augmentation problem that has performance ratio 3/2 and uses O(n2 log n) time. An O(n3) approximation algorithm with performance ratio 5/4 is presented to make a biconnected planar graph triconnected by adding edges without losing planarity.