  We introduce the game of Surrounding Cops and Robbers on a graph, as a
variant of the original game of Cops and Robbers. In contrast to the original
game in which the cops win by occupying the same vertex as the robber, they now
win by occupying each of the robber's neighbouring vertices. We denote by
$\sigma(G)$ the {\em surrounding cop number} of $G$, namely the least number of
cops required to surround a robber in the graph $G$. We present a number of
results regarding this parameter, including general bounds as well as exact
values for several classes of graphs. Particular classes of interest include
product graphs, graphs arising from combinatorial designs, and generalised
Petersen graphs.
