Examples of platonic polygonal complexes include the five regular solids, the tesselations of the Euclidean and hyperbolic planes by regular polygons, and the 2-dimensional faces of the tesselation of each Euclidean space by equal cubes. There are a lot more, and classifying them all seems to be far too difficult.I shall discuss recent joint work with T. Januszkiewicz, R. Valle and R. Vogeler in which we classify some families of these complexes.