  The purpose of this paper is to deepen the study of the Pr\"ufer
$\star$--multiplication domains, where $\star$ is a semistar operation. For
this reason, in Section 2, we introduce the $\star$--domains, as a natural
extension of the $v$--domains, where $v$ is the classical Artin's divisorial
operation. We investigate their close relation with the Pr\"ufer
$\star$--multiplication domains. In particular, in Section 3, we obtain a
characterization of Pr\"ufer $\star$--multiplication domains in terms of
$\star$--domains satisfying a variety of coherent-like conditions. In Section
4, we extend to the semistar setting the notion of $\texttt{H}$--domain
introduced by Glaz and Vasconcelos and we show, among the other results that,
in the class of the $\texttt{H}(\star)$--domains, the Pr\"ufer
$\star$--multiplication domains coincide with the $\star$--domains.
