The book deals with two sets of physical problems: 1) quantum theory of continuous measurements and 2) the theory of quantum particles in gauge and gravitational fields. These two circles of questions are united by the commonality of the mathematical apparatus (the space of paths), but are set out independently of each other. Quantum theory of continuous measurements (such as coordinate tracking or spectral measurements) is based on Feynman's path integral. This allows us to take into account the inverse influence of measuring equipment on the quantum system (the phenomenon of wave packet reduction) during continuous measurement.. The theory of gauge and gravitational fields and particles moving in these fields is based on the so-called group of paths, which generalizes the group of translations. This makes it possible, in a sense, to "reduce geometry to algebra" and makes the theory of particles in external fields similar to the theory of free particles. The connection of local and global aspects in the description of particles is discussed, the kinematics of nonlocal particles (strings, gauge monopoles) is investigated.