In this thesis by using the geometric thermodynamics methods such as Ruppeiner and Weinhold’s curvatures thermodynamic properties of some type of black holes in four and arbitrary dimension is studied. One could show that the divergence points of the Ruppeiner curvature indicates the phase transition points. Furthermore, the value of Ruppeiner curvature is a measure of interactions of the underlying statistical system and could be used to compare the results coming from different candidates of quantum gravity. Also, by studying the phase space of Reier-Nordstrom black holes we find that for detecting the right points of phase transition one must use the complete set of state space parameters and special black holes must be considered as the limits of general black holes