In addition to objects and relationships between them, groups or clusters of objects are an essential part of many real-world datasets: party affiliation in political networks, types of living organisms in the tree of life, movie genres in the internet movie database. In recent visualization methods, such group information is conveyed by explicit regions that enclose related elements. However, when in addition to fixed cluster membership, the input elements also have fixed positions in space (e.g., geo-referenced data), it becomes difficult to produce readable visualizations. In such fixed-clustering and fixed-embedding settings, some methods produce fragmented regions, while other produce contiguous (connected) regions that may contain overlaps even if the input clusters are disjoint. Both fragmented regions and unnecessary overlaps have a detrimental effect on the interpretation of the drawing. With this in mind, we propose MapSets: a visualization technique that combines the advantages of both methods, producing maps with non-fragmented and non-overlapping regions. The proposed method relies on a theoretically sound geometric algorithm which guarantees contiguity and disjointness of the regions, and also optimizes the convexity of the regions. A fully functional implementation is available in an online system and is used in a comparison with related earlier methods.