  In a first step, we establish the existence (and sometimes the uniqueness) of
solutions for a large class of quadratic backward stochastic differential
equations (QBSDEs) with continuous generator and a merely square integrable
terminal condition. Our approach is different from those existing in the
literature. Although we are focused on QBSDEs, our existence result also covers
the BSDEs with linear growth, keeping $\xi$ square integrable in both cases. As
byproduct, the existence of viscosity solutions is established for a class of
quadratic partial differential equations (QPDEs) with a square integrable
terminal datum. In a second step, we consider QBSDEs with measurable generator
for which we establish a Krylov's type a priori estimate for the solutions. We
then deduce an It\^o--Krylov's change of variable formula. This allows us to
establish various existence and uniqueness results for classes of QBSDEs with
square integrable terminal condition and sometimes a merely measurable
generator. Our results show, in particular, that neither the existence of
exponential moments of the terminal datum nor the continuity of the generator
are necessary to the existence and/or uniqueness of solutions for quadratic
BSDEs. Some comparison theorems are also established for solutions of a class
of QBSDEs.
