  We study projective varieties $X \subset \mathbb{P}^r$ of dimension $n \geq
2$, of codimension $c \geq 3$ and of degree $d \geq c + 3$ that are of maximal
sectional regularity, i.e. varieties for which the Castelnuovo-Mumford
regularity $\reg (\mathcal{C})$ of a general linear curve section is equal to
$d -c+1$, the maximal possible value (see \cite{GruLPe}). As one of the main
results we classify all varieties of maximal sectional regularity. If $X$ is a
variety of maximal sectional regularity, then either (a) it is a divisor on a
rational normal $(n+1)$-fold scroll $Y \subset \mathbb{P}^{n+3}$ or else (b)
there is an $n$-dimensional linear subspace $\mathbb{F} \subset \mathbb{P}^r$
such that $X \cap \mathbb{F} \subset \mathbb{F}$ is a hypersurface of degree
$d-c+1$. Moreover, suppose that $n = 2$ or the characteristic of the ground
field is zero. Then in case (b) we obtain a precise description of $X$ as a
birational linear projection of a rational normal $n$-fold scroll.
