  We study projective surfaces $X \subset \mathbb{P}^r$ (with $r \geq 5$) of
maximal sectional regularity and degree $d > r$, hence surfaces for which the
Castelnuovo-Mumford regularity $\reg(C)$ of a general hyperplane section curve
$C = X \cap \mathbb{P}^{r-1}$ takes the maximally possible value $d-r+3$. We
show that each of these surfaces is either a cone over a curve $C \subset
\mathbb{P}^{r-1}$ of maximal regularity or else a birational outer linear
projection of a smooth rational surface scroll $\widetilde{X} \subset
\mathbb{P}^{d+1}$. We prove that the Castelnuovo-Mumford regularity of these
surfaces satisfies the equality $\reg(X) = d-r+3$ and we compute or estimate
various of their cohomological invariants as well as their Betti numbers. We
study the the extremal variety $\mathbb{F}(X)$ of these surfaces $X$, that is
the closed union of the extremal secant lines of all smooth hyperplane section
curves of $X$. We show that $\mathbb{F}(X)$ is either a plane or that otherwise
$r =5$ and $\mathbb{F}(X)$ is a rational smooth threefold scroll $S(1,1,1)
\subset \mathbb{P}^5$.
