  In this paper we provide weighted estimates for rough operators, including
rough homogeneous singular integrals $T_\Omega$ with $\Omega\in
L^\infty(\mathbb{S}^{n-1})$ and the Bochner-Riesz multiplier at the critical
index $B_{(n-1)/2}$. More precisely, we prove qualitative and quantitative
versions of Coifman-Fefferman type inequalities and their vector-valued
extensions, weighted $A_p-A_\infty$ strong and weak type inequalities for
$1<p<\infty$, and $A_1-A_\infty$ type weak $(1,1)$ estimates. Moreover,
Fefferman-Stein type inequalities are obtained, proving in this way a
conjecture raised by the second-named author in the 90's. As a corollary, we
obtain the weighted $A_1-A_\infty$ type estimates. Finally, we study rough
homogenous singular integrals with a kernel involving a function $\Omega\in
L^q(\mathbb{S}^{n-1})$, $1<q<\infty$, and provide Fefferman-Stein inequalities
too. The arguments used for our proofs combine several tools: a recent sparse
domination result by Conde-Alonso et.al. [CACDPO], results by the first author
in [L], suitable adaptations of Rubio de Francia algorithm, the extrapolation
theorems for $A_{\infty}$ weights [CMP,CGMP] and ideas contained in previous
works by A. Seeger in [S] and D. Fan and S. Sato [FS].
