In this thesis, first we study the unique classical solution of quasi-linear second order parabolic partial differential equations (PDEs). For this, we study the existence and uniqueness of the $L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{d}) \otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k})\otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k\times d})$ valued solution of forward backward stochastic differential equations (FBSDEs) with finite horizon, the regularity property of the solution of FBSDEs and the connection between the solution of FBSDEs and the solution of quasi-linear parabolic PDEs. Then we establish their connection in the Sobolev weak sense, in order to give the weak solution of the quasi-linear parabolic PDEs. Finally, we study the unique weak solution of quasi-linear second order elliptic PDEs through the stationary solution of the FBSDEs with infinite horizon.