  We carry out a thorough survey of entropy for a large class of $p$-branes in
various dimensions. We find that the Bekenstein-Hawking entropy may be given a
simple world volume interpretation only for the non-dilatonic $p$-branes, those
with the dilaton constant throughout spacetime. The entropy of extremal
non-dilatonic $p$-branes is non-vanishing only for the solutions preserving 1/8
of the original supersymmetries. Upon toroidal compactification these reduce to
dyonic black holes in 4 and 5 dimensions. For the self-dual string in 6
dimensions, which preserves 1/4 of the original supersymmetries, the
near-extremal entropy is found to agree with a world sheet calculation, in
support of the existing literature. The remaining 3 interesting cases preserve
1/2 of the original supersymmetries. These are the self-dual 3-brane in 10
dimensions, and the 2- and 5-branes in 11 dimensions. For all of them the
scaling of the near-extremal Bekenstein-Hawking entropy with the Hawking
temperature is in agreement with a statistical description in terms of free
massless fields on the world volume.
