  We introduce a new class of Backward Stochastic Differential Equations in
which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a
random variable, but only satisfies a weak constraint of the form
$E[\Psi(Y_{T})]\ge m$, for some (possibly random) non-decreasing map $\Psi$ and
some threshold $m$. We name them \textit{BSDEs with weak terminal condition}
and obtain a representation of the minimal time $t$-values $Y_{t}$ such that
$(Y,Z)$ is a supersolution of the BSDE with weak terminal condition. It
provides a non-Markovian BSDE formulation of the PDE characterization obtained
for Markovian stochastic target problems under controlled loss in Bouchard,
Elie and Touzi \cite{BoElTo09}. We then study the main properties of this
minimal value. In particular, we analyze its continuity and convexity with
respect to the $m$-parameter appearing in the weak terminal condition, and show
how it can be related to a dual optimal control problem in Meyer form. These
last properties generalize to a non Markovian framework previous results on
quantile hedging and hedging under loss constraints obtained in F\"{o}llmer and
Leukert \cite{FoLe99,FoLe00}, and in Bouchard, Elie and Touzi \cite{BoElTo09}.
