  We study the boundedness of the Hilbert transform $H$ and the Hilbert maximal
operator $H^*$ on weighted Lorentz spaces $\Lambda^p_u(w)$. We start by giving
several necessary conditions that, in particular, lead us to the complete
characterization of the weak-type boundedness of both $H$ and $H^*$, whenever
$u\in A_1$. For the strong-type case, we also get the characterization of both
operators when $p>1$. Applications to the case of Lorentz spaces $L^{p,q}(u)$
are presented.
