The collection contains classical and modern works on dynamics in spaces of constant curvature. The problem of Kepler and its generalizations, the problems of two and three bodies, and the dynamics of a solid body in curved spaces are considered. Many of the classic works owned by B. Killing, G. Libman et al . They are not easy to read and are almost forgotten. This book is the first to publish their translation. Modern research presented in the book focuses on the issues of integrability and stochasticity, generalization of various results of classical celestial mechanics, the theory of Newtonian potential. Many results are published here for the first time. The book is intended for students and postgraduates of universities, specialists in the theory of dynamical systems, will also be of interest to historians of science.