In this thesis, we study the thermodynamics and the thermodynamic geometries of various black holes. We investigate the thermodynamics of these black holes within the context of the Weinhold and Ruppeiner thermodynamic geometries and the recently developed formalism of geometrothermodynamics (GTD). Considering the behavior of the heat capacity and the Hawking temperature, we show that Weinhold and Ruppeiner geometries cannot describe completely the thermodynamics of these black holes. In contrast, in the space of thermodynamic equilibrium states a Legendre invariant metric gives us all the information about the thermodynamics of black holes. Of course the curvature of this thermodynamic metric becomes singular at those points where, according to the analysis of the heat capacities, phase transitions occur. This means that singularities of the scalar curvature are exactly given by the phase transition points. Recently, a new solution of Enistein-anti-Maxwell theory with cosmological constant, which is called Phantom Reier-Nordstrom-AdS black hole solution (anti- Reier- Nordstrom- (A) de sitter) has been investigated. For studying the thermodynamic properties of this solution, it is necessary to express the mass (energy) in terms of the radius of the events horizon and the charge. Considering the first law of thermodynamic and mass function, it is simple to calculate intensive variables such as the Hawking temperature and the electric potential for Phantom black hole. The thermodynamic properties of Phantom Reier-Nordstrom-AdS black hole solution can be analysed in many ways. One of them is the usual study made by Davies. But there are other methods well established both physically and mathematically, which are similarly used to study the thermodynamic properties, such as that of the Geometrothermodynamics and that of Hamiltonian thermodynamics. For Phantom Reier-Nordstrom-AdS black hole, it has been verified that the phase transition points of the specific heat are incompatible with the singularities of the scalar curvature using the Geometrothermodynamic method (GTD). Thus, the Geometrodynamics method is not able to provide the same result for the analysis done by the heat capacity of anti-RN-AdS black holes and RN-AdS black hole. In this work, we propose a new formulation for the thermodynamic geometry of black holes. This formulation leads to correct relation between heat capacity and curvature singularities. A recent study shows that Geometrothermodynamics fails to represent all singularities of the specific heat for the Phantom black hole while our formulation gives correct results in this case. We also investigated a large clan of black holes where in all of them the singularity of specific heat corresponds to that of scalar curvature. We conclude that our method can be used as a correct and simple formulation of thermodynamic geometry and is also applicable to physics of black holes. Keywords: Thermodynamic geometry, The specific heat, Scalar curvature