  Let $\ast$ be a star operation on an integral domain $D$. Let $\f(D)$ be the
set of all nonzero finitely generated fractional ideals of $D$. Call $D$ a
$\ast$--Pr\"ufer (respectively, $(\ast, v)$--Pr\"ufer) domain if
$(FF^{-1})^{\ast}=D$ (respectively, $(F^vF^{-1})^{\ast}=D$) for all $F\in
\f(D)$. We establish that $\ast$--Pr\"ufer domains (and $(\ast, v)$--Pr\"ufer
domains) for various star operations $\ast $ span a major portion of the known
generalizations of Pr\"{u}fer domains inside the class of $v$--domains. We also
use Theorem 6.6 of the Larsen and McCarthy book [Multiplicative Theory of
Ideals, Academic Press, New York--London, 1971], which gives several equivalent
conditions for an integral domain to be a Pr\"ufer domain, as a model, and we
show which statements of that theorem on Pr\"ufer domains can be generalized in
a natural way and proved for $\ast$--Pr\"ufer domains, and which cannot be. We
also show that in a $\ast $--Pr\"ufer domain, each pair of $\ast $-invertible
$\ast $-ideals admits a GCD in the set of $\ast $-invertible $\ast $-ideals,
obtaining a remarkable generalization of a property holding for the "classical"
class of Pr\"ufer $v$--multiplication domains. We also link $D$ being $\ast
$--Pr\"ufer (or $(\ast, v)$--Pr\"ufer) with the group Inv$^{\ast}(D)$ of $\ast
$-invertible $\ast $-ideals (under $\ast$-multiplication) being
lattice-ordered.
