  We present a comprehensive theory on the well-posedness of a one-dimensional
nonlinear backward stochastic differential equation (1D BSDE for short), where
the generator $g$ has a one-sided linear/super-linear growth in the first
unknown variable $y$ and an at most quadratic growth in the second unknown
variable $z$. We first establish several existence theorems and comparison
theorems with the test function method and the a priori estimate technique, and
then immediately give several existence and uniqueness results. We also
overview relevant known results and introduce some practical applications of
our theoretical results. Finally, we list some open problems on the
well-posedness of 1D BSDEs.
