We will revise the basic definitions and facts proved in the last talk. We start by proving the Fundamental observation of Geometric group theory and then go on to $\delta$-hyperbolic spaces. We will define the notion of quasi-geodesics and prove some of their properties. We will then prove the fact that $\delta$-hyperbolicity is preserved under quasi-isometries. If time permits, we define Gromov boundary and state some of its properties.