  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
