The current data on scattering and annihilation of nonrelativistic antinucleons are reviewed. A description is given of the theoretical ideas on the relation between the nucleon-antinucleon and nucleon-nucleon interactions, which make it possible to organize the experimental data on the elastic scattering of nucleons into a picture of the forces which act between the nucleon and antinucleon. The main conclusion from the existing experimental facts and theoretical considerations is that we should expect a strong attraction due to the exchange of a neutral vector meson (the omega meson) in the nucleon-antinucleon system at distances of the order of 1--2 F. Calculations show that this attraction leads to the existence of bound and resonant nuclear-like states in the nucleon-antinucleon system, which should manifest themselves experimentally as heavy mesons of mass close to the mass of two nucleons (quasinuclear mesons). The lifetime of such mesons is determined by the probability of the annihilation process (which occurs at small distances of the order of the nucleon Compton wavelength and consequently has a small effect on the production of the quasinuclear state). A distinctive feature of quasinuclear mesons is their comparatively large probability of decay (virtual or real) into the nucleon-antinucleon channel. For this reason, the production of quasinuclear mesons should be seen in collisions of antinucleons with protons and nuclei. Another characteristic consequence of the existence of quasinuclear mesons is an anomalous energy dependence of the cross section for annihilation of slow antinucleons. The theoretical predictions are in agreement with the latest experimental results, which are discussed in detail in the paper.