  A pebbling move on a graph removes two pebbles at a vertex and adds one
pebble at an adjacent vertex. Rubbling is a version of pebbling where an
additional move is allowed. In this new move, one pebble each is removed at
vertices $v$ and $w$ adjacent to a vertex $u$, and an extra pebble is added at
vertex $u$. A vertex is reachable from a pebble distribution if it is possible
to move a pebble to that vertex using rubbling moves. The rubbling number is
the smallest number $m$ needed to guarantee that any vertex is reachable from
any pebble distribution of $m$ pebbles. The optimal rubbling number is the
smallest number $m$ needed to guarantee a pebble distribution of $m$ pebbles
from which any vertex is reachable. We give bounds for rubbling and optimal
rubbling numbers. In particular, we find an upper bound for the rubbling number
of $n$-vertex, diameter $d$ graphs, and estimates for the maximum rubbling
number of diameter 2 graphs. We also give a sharp upper bound for the optimal
rubbling number, and sharp upper and lower bounds in terms of the diameter.
