  We study the arithmetic properties of projective varieties of almost minimal
degree, that is of non-degenerate irreducible projective varieties whose degree
exceeds the codimension by precisely 2. We notably show, that such a variety $X
\subset {\mathbb P}^r$ is either arithmetically normal (and arithmetically
Gorenstein) or a projection of a variety of minimal degree $\tilde {X} \subset
{\mathbb P}^{r + 1}$ from an appropriate point $p \in {\mathbb P}^{r + 1}
\setminus \tilde {X}$. We focus on the latter situation and study $X$ by means
of the projection $\tilde {X} \to X$.
  If $X$ is not arithmetically Cohen-Macaulay, the homogeneous coordinate ring
$B$ of the projecting variety $\tilde {X}$ is the endomorphism ring of the
canonical module $K(A)$ of the homogeneous coordinate ring $A$ of $X.$ If $X$
is non-normal and is maximally Del Pezzo, that is arithmetically Cohen-Macaulay
but not arithmetically normal $B$ is just the graded integral closure of $A.$
It turns out, that the geometry of the projection $\tilde {X} \to X$ is
governed by the arithmetic depth of $X$ in any case.
  We study in particular the case in which the projecting variety $\tilde {X}
\subset {\mathbb P}^{r + 1}$ is a cone (over a) rational normal scroll. In this
case $X$ is contained in a variety of minimal degree $Y \subset {\mathbb P}^r$
such that $\codim_Y(X) = 1$. We use this to approximate the Betti numbers of
$X$.
  In addition we present several examples to illustrate our results and we draw
some of the links to Fujita's classification of polarized varieties of $\Delta
$-genus 1.
