  We study the existence and structure of $d$-polytopes for which the number
$f_1$ of edges is small compared to the number $f_0$ of vertices. Our results
are more elegantly expressed in terms of the excess degree of the polytope,
defined as $2f_1-df_0$. We show that the excess degree of a $d$-polytope cannot
lie in the range $[d+3,2d-7]$, complementing the known result that values in
the range $[1,d-3]$ are impossible. In particular, many pairs $(f_0,f_1)$ are
not realised by any polytope. For $d$-polytopes with excess degree $d-2$,
strong structural results are known; we establish comparable results for excess
degrees $d$, $d+2$, and $2d-6$. Frequently, in polytopes with low excess
degree, say at most $2d-6$, the nonsimple vertices all have the same degree and
they form either a face or a missing face. We show that excess degree $d+1$ is
possible only for $d=3,5$, or $7$, complementing the known result that an
excess degree $d-1$ is possible only for $d=3$ or $5$.
