  We investigate generalizations of pebbling numbers and of Graham's pebbling
conjecture that pi(GxH) <= pi(G)pi(H), where pi(G) is the pebbling number of
the graph G. We develop new machinery to attack the conjecture, which is now
twenty years old. We show that certain conjectures imply others that initially
appear stronger. We also find counterexamples that show that Sjostrand's
theorem on cover pebbling does not apply if we allow the cost of transferring a
pebble from one vertex to an adjacent vertex to depend on the edge, and we
describe an alternate pebbling number for which Graham's conjecture is
demonstrably false.
