In this article a critical analysis is given of basic geometric concepts as applied to the physics of the microcosmos. The article begins with an outline of geometry in macroscopic physics. This part is based on the classical papers of H. Poincare, A. Einstein, and A. A. Fridman. An indication of possible limitations on the concept of a point event associated with limiting densities of matter represents new material. Subsequently localization of elementary particles is discussed and limits of possible accuracy are indicated. In the article it is emphasized that the logical structure of local quantum field theory presupposes the existence of particles of arbitrarily large mass. The existence of ``maximons''–particles of a limiting, but finite mass–strong gravitation (collapse of particles), instability of particles with respect to decay due to the weak interaction (this interaction can become strong for very heavy particles and can lead to lead to total instability of such a particle). In the conclusion of the article two features of a nonlocal field theory are considered. In this theory the coordinates of a point event are operators, while momentum space remains a numerical space (either curved or flat). The first variant presupposes commutation conditions for the coordinates of a point (the Snyder--Kadyshevskii theory), while the second variant presupposes anticommutation conditions (the author's theory).