  We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump
condition in the scale of log bumps and certain loglog bumps, then Haar shifts
map $L^p(v)$ into $L^p(u)$ with a constant quadratic in the complexity of the
shift. This in turn implies the two weight boundedness for all
Calder\'on-Zygmund operators. This gives a partial answer to a long-standing
conjecture. We also give a partial result for a related conjecture for
weak-type inequalities. To prove our main results we combine several different
approaches to these problems; in particular we use many of the ideas developed
to prove the $A_2$ conjecture. As a byproduct of our work we also disprove a
conjecture by Muckenhoupt and Wheeden on weak-type inequalities for the Hilbert
transform. This is closely related to the recent counterexamples of Reguera,
Scurry and Thiele.
