  A pebbling move on a graph consists of taking two pebbles off of one vertex
and placing one pebble on an adjacent vertex. In the traditional pebbling
problem we try to reach a specified vertex of the graph by a sequence of
pebbling moves. In this paper we investigate the case when every vertex of the
graph must end up with at least one pebble after a series of pebbling moves.
The cover pebbling number of a graph is the minimum number of pebbles such that
however the pebbles are initially placed on the vertices of the graph we can
eventually put a pebble on every vertex simultaneously. We find the cover
pebbling numbers of trees and some other graphs. We also consider the more
general problem where (possibly different) given numbers of pebbles are
required for the vertices.
