This thesis is going to give a gentle introduction to Mean Field Games. It aims to produce a coherent text beginning for simple notions of deterministic control theory progressively to current Mean Field Games theory. The framework gradually extended form single agent stochastic control problems to multi agent stochastic differential mean field games. The concept of Nash Equilibrium is introduced to define a solution of the mean field game. To achieve considerable simplifications the number of agents goes to infinity and formulate this problem on the basis of McKean-Vlasov theory for interacting particle systems. Furthermore, the problem at infinity is being solved by a variation of the Stochastic Maximum Principle and Forward Backward Stochastic Differential Equations. To elaborate more the Aiyagari macroeconomic model in continuous time is presented using MFGs techniques