  For a graph $G$ whose degree sequence is $d_{1},..., d_{n}$, and for a
positive integer $p$, let $e_{p}(G)=\sum_{i=1}^{n}d_{i}^{p}$. For a fixed graph
$H$, let $t_{p}(n,H)$ denote the maximum value of $e_{p}(G)$ taken over all
graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly,
$t_{1}(n,H)$ is twice the Tur\'{a}n number of $H$. In this paper we consider
the case $p>1$. For some graphs $H$ we obtain exact results, for some others we
can obtain asymptotically tight upper and lower bounds, and many interesting
cases remain open.
