Fixed point theory provides a useful tool for solving problems arising in various branches of mathematical analysis and its related disciplines. This theory deals with the investigation of less restrictive conditions on mappings that guarantee the of existence of their fixed point and study of conditions which assure uniqueness of fixed point. On the other hand, though best approximation theorems ensure the existence of approximation solutions, such results need not yield optimal solutions. But, best proximity point theorems provide sufficient conditions that assure the existence of approximate solutions which are optimal as well. In this thesis, we are mainly concerned convex type contractions, α-η-Weakly Zamfirescu contraction and α-η-Ciri´c strong al- ´ most contraction. We obtained related fixed point and approximate fixed theorems in the framework of complete G-metric space which are more general than ordinary metric spaces. We also obtained coincidence best proximity point results of Fg-weak contraction mapping and (ϕ, θ, α, g)-contraction mapping in partially ordered metric spaces and in G-metric spaces, respectively. Best proximity points for proximal cyclic contraction of Perov type, quasi contraction, Perov-Ciric quasi contraction and Perov-Fisher quasi contraction in the setup of complete cone metric space are also obtained.