This paper presents the final results of the ICDAR 2021 Competition on Historical Map Segmentation (MapSeg), encouraging research on a series of historical atlases of Paris, France, drawn at 1/5000 scale between 1894 and 1937. The competition featured three tasks, awarded separately. Task 1 consists in detecting building blocks and was won by the L3IRIS team using a DenseNet-121 network trained in a weakly supervised fashion. This task is evaluated on 3 large images containing hundreds of shapes to detect. Task 2 consists in segmenting map content from the larger map sheet, and was won by the UWB team using a U-Net-like FCN combined with a binarization method to increase detection edge accuracy. Task 3 consists in locating intersection points of geo-referencing lines, and was also won by the UWB team who used a dedicated pipeline combining binarization, line detection with Hough transform, candidate filtering, and template matching for intersection refinement. Tasks 2 and 3 are evaluated on 95 map sheets with complex content. Dataset, evaluation tools and results are available under permissive licensing at https://icdar21-mapseg.github.io/.