  This paper is concerned with initial-boundary-value problems (IBVPs) for a
class of nonlinear Schr\"odinger equations posed either on a half line
$\mathbb{R}^+$ or on a bounded interval $(0, L)$ with nonhomogeneous boundary
conditions. For any $s$ with $0\leq s < 5/2$ and $s \not = 3/2$, it is shown
that the relevant IBVPs are locally well-posed if the initial data lie in the
$L^2$--based Sobolev spaces $H^s(\mathbb{R}^+) $ in the case of the half line
and in $H^s (0, L)$ on a bounded interval, provided the boundary data are
selected from $H^{(2s+1)/4}_{loc} (\mathbb{R}^+)$ and $H^{(s+ 1) /2}_{loc}
(\mathbb{R}^+)$, respectively. (For $s > \frac12$, compatibility between the
initial and boundary conditions is also needed.) Global well-posedness is also
discussed when $s \ge 1$. From the point of view of the well-posedness theory,
the results obtained reveal a significant difference between the IBVP posed on
$\mathbb{R}^+$ and the IBVP posed on $(0,L)$. The former is reminiscent of the
theory for the pure initial-value problem (IVP) for these Schr\"odinger
equations posed on the whole line $\mathbb{R}$ while the theory on a bounded
interval looks more like that othe pure IVP posed on a periodic domain. In
particular, the regularity demanded of the boundary data for the IBVP on
$\mathbb{R}^+$ is consistent with the temporal trace results that obtain for
solutions of the pure IVP on $\mathbb{R}$, while the slightly higher regularity
of boundary data for the IBVP on $(0, L)$ resembles what is found for temporal
traces of spatially periodic solutions.
