In graph theory, questions related to planarity always played an important role. The main subject of this chapter is the following problem, which is denoted here as Maximum Weight Planar Subgraph: given a graph G with a nonnegative weight defined for each edge, find a planar subgraph of G of maximum weight, in which the weight of a subgraph is simply the sum of the weights of the edges in the subgraph. Its unweighted version, denoted as Maximum Planar Subgraph, consists of: given a simple graph G, find a planar subgraph of G with the maximum number of edges.