Relativistic velocity is a kinematic feature of micro objects (elementary particles). Their application to macro objects (stars, planets, asteroids, neutron stars, and stellar-mass black holes) is currently under scientific discussion. This potential was recognized after Warren Brown discovered hypervelocity stars (HVSs) at the beginning of the 21st century. Jack Hills predicted these stars in 1988 due to the dynamical capture of a binary star by the central supermassive black hole (SMBH). The acceleration mechanism due to momentum exchange in the classical three-body problem provides the kinetic resource for HVS formation by the gravitational capture of the remaining component. The present threshold of the anomalous stellar kinematics exceeds similar to 1700 km s(-1) and can be reproduced by some mechanisms as alternatives to Hills's scenario. HVSs can arise due to the collisional evolution of stellar clusters, supernova explosions in close binary stars, the orbital instability of triple stars, stellar captures from other galaxies, etc. Scenarios with the participation of black holes with masses ranging from stellar values to several billion solar masses are the most promising for the generation of anomalously high stellar velocities. Hills's scenario has a special place in HVS studies, because, being based on the accidental capture of a binary star by the SMBH, it does not relate to the problem of the Galactic Center population. This scenario predicts self-consistent statistics of HVSs and captured stars which may be identified with S-stars. The discovery of S-stars played an essential role in studies of the Galactic Center; their dynamics have independently provided incontestable proof of the SMBH's existence. This review briefly discusses the history of the discovery and investigation of HVSs and S-stars, provides an account of their observational statistics, and describes their modeling methods in the classical three-body and N body problems. We study the limits of the effective acceleration of stars in the classical Hills scenario and the modified mechanism that allows a change of one of the binary components to another SMBH. The acceleration acquired by the star in a mutual field of two SMBHs can produce stars with relativistic velocities (1/2 c -2/3 c). Using a self-consistent probabilistic model combining the classical and modified Hills scenarios, we predict the formation probability of HVSs in the Galaxy and of extragalactic stars with relativistic velocities. We discuss the prospects of searches for stars and asteroids with relativistic velocities by future space missions and using new knowledge about the Universe.
