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.