Complex hyperbolic 2-space can be seen as the unit ball in C^2 (the 2-dimensional complex vector space) with the Bergman metric. This is a complex counterpart of the Poincare disk model of real hyperbolic plane. In this talk, I will introduce the geometry of complex hyperbolic 2-space. We will start with basic definitions such as isometries and the boundary at infinity along with other geometric notions. Then we will move to a research survey regarding the action of discrete groups of isometries.