Fixed point theory is a charming compound of analysis (applied and pure), geometry and topology. It is one of the most powerful and productive tools in the nonlinear analysis and it can be considered the Kernel of the nonlinear analysis. We also used the fixed point theory in biology, chemistry, engineering etc. It also has many fruitful applications in control theory, game theory, economics and many other areas. In the existence theory of differential equations partial differential equations and numerical methods fixed point theorems are also used. Moreover, the applications of fixed point theory increase enormously in various branches of sciences and engineering. Fixed points are the points which remain invariant under a map/transformation. Fixed points tell us about the parts of the space which are pinned in plane not moved by the transformation. The fixed points of a transformation restrict the motion of the space under some restrictions. A map can have one fixed point, two fixed point, infinitely many fixed point and No fixed point. It is not necessary that all functions have fixed point. We note that translation mapping has no fixed point. For example, if f(x) = x +3, clearly it is translation mapping. Now the question arise what type of problems have the fixed point. Also, we state the results which give us the guarantee of existence of fixed point. In this project, I presented the notation of 2-metric space, complete 2-metric space, 2-normed space, complete 2-normed space, expansion mappings in 2-metric spaces, fixed point theorems in 2-metric space and in 2-Banach spaces provided with some definitions and examples.