Abstarct: In this thesis‎, ‎the theory of best approximation and the best proximity point which are the generalization of fixed point theory for non-self mapping on metric and metric like spaces (2-norm‎, ‎2-metric‎, ‎and modular Metric‎, .‎..)‎ ‎to be studied‎. ‎we review the 2-metric spaces and define and extend some of the concepts for this type of spaces‎. ‎‎ ‎W‎e are investigating the 2-noemed spaces and then expanding the theory of best approximation in this type of spaces too‎. We have completed the theory of the best proximity point in complete metric spaces and complete 2-metric spaces‎, ‎and by presenting the definition of ϕ -‎contractions mappings and P‎ ‎-properties,‎ ‎we present new results in this theory‎.