Map grammars are discussed first of all in their relationships to graph grammars. Maps are more strictly specified and better implementable structures than planar graphs. While graphs consist of vertices and edges, maps can be defined as sets of vertices, edges and regions. After the original introduction of sequential map grammars, most of the recent work has been on parallel map generating systems. The main types of these are: (1) binary fission/fusion systems with labeling and interactions of the regions, (2) the map interpretations of parallel graph grammars (such as propagating graph OL-systems) with node (region) labeling, and (3) the edgelabel-controlled binary propagating map OL-systems (BPMOL-systems). Of the latter systems various classes are defined: with single or double edge labels, and with edge insertion controlled by cicular words or by markers. Properties and applicability to cell division patterns of these families of systems are compared.