  We prove several sharp weighted norm inequalities for commutators of
classical operators in harmonic analysis. We find sufficient $A_p$-bump
conditions on pairs of weights $(u,v)$ such that $[b,T]$, $b\in BMO$ and $T$ a
singular integral operator (such as the Hilbert or Riesz transforms), maps
$L^p(v)$ into $L^p(u)$. Because of the added degree of singularity, the
commutators require a "double log bump" as opposed to that of singular
integrals, which only require single log bumps. For the fractional integral
operator $I_\al$ we find the sharp one-weight bound on $[b,I_\al]$, $b\in BMO$,
in terms of the $A_{p,q}$ constant of the weight. We also prove sharp
two-weight bounds for $[b,I_\al]$ analogous to those of singular integrals. We
prove two-weight weak-type inequalities for $[b,T]$ and $[b,I_\al]$ for pairs
of factored weights. Finally we construct several examples showing our bounds
are sharp.
