In many practical applications with a graph structure, there may exist crossing between edges which is not allowed in a crisp planar graph. Considering that graphic structures can be used to describe the crossing of edges in an m-polar fuzzy multi-graph with certain amount of m-polar fuzzy planarity value, first of all the notion of m-polar fuzzy multi-set is introduced. Then m-polar fuzzy multi-graphs, m-polar fuzzy planar graphs and m-polar fuzzy strong edges are defined. Several properties of m-polar fuzzy planar graphs are also studied.