  We consider the Schr\"odinger equation on a half space in any dimension with
a class of nonhomogeneous boundary conditions including Dirichlet, Neuman and
the so-called transparent boundary conditions. Building upon recent local in
time Strichartz estimates (for Dirichlet boundary conditions), we obtain global
Strichartz estimates for initial data in $H^s,\ 0\leq s\leq 2$ and boundary
data in a natural space $\mathcal{H}^s$. For $s\geq 1/2$, the issue of
compatibility conditions requires a thorough analysis of the $\mathcal{H}^s$
space. As an application we solve nonlinear Schr\"odinger equations and
construct global asymptotically linear solutions for small data. A discussion
is included on the appropriate notion of scattering in this framework, and the
optimality of the $\mathcal{H}^s$ space.
