  We investigate the Hardy-Schr\"odinger operator $L_\gamma=-\Delta
-\frac{\gamma}{|x|^2}$ on domains $\Omega\subset\rn$, whose boundary contain
the singularity $0$. The situation is quite different from the well-studied
case when $0$ is in the interior of $\Omega$. For one, if $0\in\Omega$, then
$L_\gamma$ is positive if and only if $\gamma<\frac{(n-2)^2}{4}$, while if
$0\in\partial\Omega$ the operator $L_{\gamma}$ could be positive for larger
value of $\gamma$, potentially reaching the maximal constant $\frac{n^2}{4}$ on
convex domains.
  We prove optimal regularity and a Hopf-type Lemma for variational solutions
of corresponding linear Dirichlet boundary value problems of the form
$L_{\gamma} u=a(x)u$, but also for non-linear equations including $L_{_\gamma}
u=\frac{|u|^{\crits-2}u}{|x|^s}$, where $\gamma <\frac{n^2}{4}$, $s\in [0,2)$
and $\crits:=\frac{2(n-s)}{n-2}$ is the critical Hardy-Sobolev exponent. We
also provide a Harnack inequality and a complete description of the profile of
all positive solutions --variational or not-- of the corresponding linear
equation on the punctured domain. The value $\gamma=\frac{n^2-1}{4}$ turned out
to be another critical threshold for the operator $L_\gamma$, and our analysis
yields a corresponding notion of "Hardy singular boundary-mass"
$m_\gamma(\Omega)$ of a domain $\Omega$ having $0\in \partial \Omega$, which
could be defined whenever $\frac{n^2-1}{4}<\gamma<\frac{n^2}{4}$.
