Efficient algorithms for finding maximum flow in planar networks are presented. These algorithms take advantage of the planarity and are superior to the most efficient algorithms to date. If the source and the terminal are on the same face, an algorithm of Berge is improved and its time complexity is reduced to $O(n\log n)$. In the general case, for a given $D > 0$ a flow of value D is found if one exists; otherwise, it is indicated that no such flow exists. This algorithm requires $O(n^2 \log n)$ time. If the network is undirected a minimum cut may be found in $O(n^2 \log n)$ time. All algorithms require $O(n)$ space.