Given a Calder\'on-Zygmund operator $T$ and two weights $u$ and $v$, we study sufficient conditions for this operator to be bounded from the space $L^p(u)$ to $L^p(v)$. We also study sharp bounds for the corresponding norm. Further, we study a question about conditions for boundedness of all Calder\'on-Zygmund operators from $L^p(u)$ to $L^p(v)$. We do it in the Euclidian setting and in metric spaces. Finally, we study the limiting case $p=1$ and the case $u=v$, when the operator has a chanse to be weakly bounded, i.e. bounded from the space $L^1(u)$ to the space $L^{1, \infty}(u)$.In particular, we disprove the ``$A_1$ conjecture'', prove the ``$A_2$ conjecture'' in the metric space setting, prove the ``bump conjecture'' for $p=2$; moreover, we state the ``separated bump conjecture'' and prove it in several particular cases.
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