  Our primary result is that a demi-normal quasi-projective variety can be
embedded in a demi-normal projective variety. Recall that a demi-normal variety
$X$ is a variety with properties $S_2$, $G_1$, and seminormality. Equivalently,
$X$ has Serre's $S_2$ property and there is an open subvariety $U$ with
complement of codimension at least 2 in $X$, such that the only singularities
of $U$ are (analytically) double normal crossings. The term demi-normal was
coined by Koll\'ar in \cite{Kol13}. As a consequence of this embedding theorem,
we prove a semi-smooth Grauert-Riemenschneider vanishing theorem for
quasi-projective varieties, the projective case having been settled in
\cite{Berq14}. The original form of this vanishing result appears in
\cite{GR70}. We prove an analogous result for semi-rational singularities. The
definition of semi-rationality requires that the choice of a semi-resolution is
immaterial. This also has been established in the projective case in
\cite{Berq14}. The analogous result for quasi-projective varieties is settled
here. Semi-rational surface singularities have also been studied in
\cite{vS87}.
