  The theory of backward SDEs extends the predictable representation property
of Brownian motion to the nonlinear framework, thus providing a path-dependent
analog of fully nonlinear parabolic PDEs. In this paper, we consider backward
SDEs, their reflected version, and their second-order extension, in the context
where the final data and the generator satisfy $L^1$-type of integrability
condition. Our main objective is to provide the corresponding existence and
uniqueness results for general Lipschitz generators. The uniqueness holds in
the so-called Doob class of processes, simultaneously under an appropriate
class of measures. We emphasize that the previous literature only deals with
backward SDEs, and requires either that the generator is separable in $(y,z)$,
see Peng [Pen97], or strictly sublinear in the gradient variable $z$, see
[BDHPS03], or that the final data satisfies an $L\ln L$-integrability
condition, see [HT18]. We by-pass these conditions by defining
$L^1$-integrability under the nonlinear expectation operator induced by the
previously mentioned class of measures.
