Geometry based on the five postulates proposed by Euclid was considered the only geometry possible for more than two millennia. It remained unchallenged until the early 19th century, when mathematicians Lobachevsky, Bolyai and Gauss, independently discovered that by modifying the Parallel postulate, they were able to develop an axiomatic system that significantly differed from the Euclidean geometry, but was equally consistent. At the beginning of the thesis, we follow a concise historical course of events that preceded the revolutionary breakthrough of the hyperbolic geometry. Afterwards, we state results unique to this axiomatic system, which mostly disagree with our intuitive experience. The underlying subject of the study are parallels, and we classify them according to their characteristics on divergent and asymptotic parallels. Then we introduce the most common models of hyperbolic geometry, on which we demonstrate the previously proven properties of two kinds of parallels, and various abstract concepts of hyperbolic geometry. The following models are observed: the pseudosphere, Beltrami-Klein disk, Poincaré disk, Poincaré half-plane and hemisphere. Using the hemisphere model, we establish transformations between the Beltrami-Klein and Poincaré models. Finally, we briefly discuss the connection of the hyperbolic geometry with the physical world and its role in Einstien’s theory of general relativity.