In the `Covering' game on a graph, a robber and a set of cops play alternately, with the cops
each moving to a vertex at distance at most 1 from their current vertex and the robber moving to a vertex
at distance at most 2 from his current vertex. The cops win if, after every one of their moves, there is
always a cop at the same vertex as the robber. How few cops are needed? We investigate this problem
for the two-dimensional grid. There are applications to the game of `Catching a Fast Robber', and our
work answers questions of Bollobas and Leader and of Balister, Bollobas, Narayanan and Shaw.