It has been established that the thermodynamic properties of black holes depend crucially on the choice of statistical ensemble in contrast to conventional thermodynamic system. It is generally thought that the local stability of a black hole is mainly determined by its heat capacity. Negative heat capacity usually gives a thermodynamically unstable system and the positive one implies a stable one. The points, where the heat capacity diverges, are usually consistent with the points, where the second-order phase transition takes place. The properties of a thermodynamic system can also be studied with the ideas of geometry. Weinhold firstly introduced the geometrical concept into the thermodynamics. Few years later, Ruppeiner introduced another metric. In particular, it was found that the Ruppeiner geometry carries the information of phase structure of a thermodynamic system. The Ruppeiner geometry lead to a zero curvature, which means there exist no phase transition points. A Legendre invariant metric was introduced by them, which could reproduce correctly the behavior of the thermodynamic interactions and phase transitions for black hole. Liu et. al. introduced a new thermodynamical metric and demonstrate with various black holes in ADS Background that the divergence of the specific heat corresponds to the curvature singularity. In this thesis we study thermodynamic of some type of black holes on the state space . Firstly, we obtained thermodynamic curvature by using the Quevedo Metric for Kerr- Newman and Kerr-ADS black hole. The thermodynamic stability is also studied. Under the assumption of an invariable cosmological constant, the integral mass formulas in the case of rotating black holes, in ADS and dS backgrounds can never be satisfied any more. Thus we study phase structure of BTZ black hole by considering the cosmological constant as a variable. We discussed the local thermodynamics stability of BTZ black hole through the heat capacity. Also we examine the correspondence between curvature singularities and phase transition point by Ruppeiner, Quevedo and Liu metrics. Finally we found that only the Quevedo metric failed to predict the behavior of phase space completely. It should be noted that this result is also valid for Kerr- DS sitter Space-time. Keywords: Thermodynamics, Black Holes, Geometry, Phase Transition, Heat Capacity