We shall divide the topic into four parts:

Classical hyperbolic geometry. We shall recall some metric aspects of a hyperbolic space as one-sheeted hyperboloid including geodesics, trigonometry, isometries, and define the ideal boundary. Then we shall observe them via conformal models in the ball and in the upper half-space.

Hyperbolic manifolds. We shall show totally geodesic and totally umbilical hypersurfaces in a hyperbolic space. Moreover, we use de Sitter space for study geometry of them, including some properties of foliations. Finally, we shall show hyperbolic surfaces and 3-manifolds through discontinuous groups of hyperbolic isometries.

Gromov hyperbolicity. We shall start from CAT(0) space as the most natural extension of comparison methods in differential geometry. Then we study hypebolic spaces in the sense of Gromov with their properties focusing on the ideal boundary. As an application we look at finitely generated groups as metric space on which geometric group theory is founded.

Complex hyperbolic geometry and other generalizations. In this part we shall describe a complex hyperbolic space over complex numer and observe similaraties and differences with the real case. Then we mention on other generalizations like quaternionic and infinite-dimensional ones.