Abstarct: In the thesis, we introduce different black holes and study their thermodynamics of processing and also introduce the laws of thermodynamics for black holes. In the following,we will introduce concepts such as Ruppenier’s thermodynamic geometry, and apply this approach to the use of phase study of black holes. Since the one-to-one matching between singularities of heat capacity and singularities of geometric scalar in Ruppenier approach,we also introduce the new approach in this field. In this new geometric approach, called the new formalism of thermodynamic (NTG) geometry, the correct relationship between the phase transition for heat capacity and the singularities in scalar curvatures is established. Therefore, we have used this new approach in the remainder of this thesis. Then, critical exponents and critical amplitude of thermodynamic intrinsic and extrinsic curvatures of the large-small black hole phase for the charged AdS black hole were studied. At the critical point, we find that the normalized intrinsic curvature RN and the extrinsic curvature KN represent the critical values 2 and 1, respectively. Accordingly, the general domains RNt2 and KNt with the temperature parameter t = T/Tc − 1 where Tc is the critical temperature value or critical temperature. Near the critical point, we find that the critical domain RNt2 and KNt when t → 0+, will be equal to −1/2, while in the range t → 0−. These two parameters are RNt2 ≈ −1/8 and KNt ≈ −1/4. These results apply not only to four-dimensional charged AdS black holes, but also to larger dimensions. Then, at the end, we study the thermodynamic curvatures associated with NTG geome tryfor the five-dimensional Gauss-bonnet neutral black hole in the AdS space. Our results show that the intrinsic and extrinsic curvature gets the same universal amplitudes obtained for charged AdS black holes so that they are independent of the Gauss-Bonnetcoupling values.