  In this paper, we study the existence and uniqueness of solutions to
quadratic Backward Stochastic Differential Equations (QBSDEs for short) with
rough driver and square integrable terminal condition. The main idea consists
in using both Doss-Sussman and Zvonkin type transformations. As an application
we study connection between QBSDEs and quadratic PDEs with rough drivers. We
also obtain Backward Doubly SDEs and QBSDEs driven by Fractional Brownian with
Hurst parameter greater than $\frac{1}{4}$ as particular cases of our QBSDEs
with rough drivers. A probabilistic representation of a class of rough
quadratic PDE is also proved.
