This paper presents the methods that have participated in the SHREC 2022 track on the fitting and recognition of simple geometric primitives on point clouds. As simple primitives we mean the classical surface primitives derived from constructive solid geometry, i.e., planes, spheres, cylinders, cones and tori. The aim of the track is to evaluate the quality of automatic algorithms for fitting and recognizing geometric primitives on point clouds. Specifically, the goal is to identify, for each point cloud, its primitive type and some geometric descriptors. For this purpose, we created a synthetic dataset, divided into a training set and a test set, containing segments perturbed with different kinds of point cloud artifacts. Among the six participants to this track, two are based on direct methods, while four are either fully based on deep learning or combine direct and neural approaches. The performance of the methods is evaluated using various classification and approximation measures.