  Globally hyperbolic spacetimes with timelike boundary $(\overline{M} = M \cup
\partial M, g)$ are the natural class of spacetimes where regular boundary
conditions (eventually asymptotic, if $\overline{M}$ is obtained by means of a
conformal embedding) can be posed. $\partial M$ represents the naked
singularities and can be identified with a part of the intrinsic causal
boundary. Apart from general properties of $\partial M$, the splitting of any
globally hyperbolic $(\overline{M},g)$ as an orthogonal product ${\mathbb
R}\times \bar{\Sigma}$ with Cauchy slices with boundary $\{t\}\times
\bar{\Sigma}$ is proved. This is obtained by constructing a Cauchy temporal
function $\tau$ with gradient $\nabla \tau$ tangent to $\partial M$ on the
boundary. To construct such a $\tau$, results on stability of both, global
hyperbolicity and Cauchy temporal functions are obtained. Apart from having
their own interest, these results allow us to circumvent technical difficulties
introduced by $\partial M$. As a consequence, the interior $M$ both, splits
orthogonally and can be embedded isometrically in ${\mathbb L}^N$, extending so
properties of globally spacetimes without boundary to a class of causally
continuous ones.
