  This paper aims at solving one-dimensional backward stochastic differential
equations (BSDEs) under weaker assumptions. We establish general existence,
uniqueness, and comparison results for bounded solutions, $L^p (p>1)$ solutions
and $L^1$ solutions of the BSDEs. The time horizon is allowed to be finite or
infinite, and the generator $g$ is allowed to have a general growth in $y$ and
a quadratic growth in $z$. As compensation, the generator $g$ needs to satisfy
a kind of one-sided linear or super-linear growth condition in $y$, instead of
the monotonicity condition in $y$ as is usually done. Many of our results
improve virtually some known results, even though for the case of the finite
time horizon and the case of the $L^2$ solution.
