  We present a general analysis of the thermodynamics of spinning black
p-branes of string and M-theory. This is carried out both for the
asymptotically-flat and near-horizon case, with emphasis on the latter. In
particular, we use the conjectured correspondence between the near-horizon
brane solutions and field theories with 16 supercharges in various dimensions
to describe the thermodynamic behavior of these field theories in the presence
of voltages under the R-symmetry. Boundaries of stability are computed for all
spinning branes both in the grand canonical and canonical ensemble, and the
effect of multiple angular momenta is considered. A recently proposed
regularization of the field theory is used to compute the corresponding
boundaries of stability at weak coupling. For the D2, D3, D4, M2 and M5-branes
the critical values of Omega/T in the weak and strong coupling limit are
remarkably close. Finally, we also show that for the spinning D3-brane the tree
level R^4 correction supports the conjecture of a smooth interpolating function
between the free energy at weak and strong coupling.
