  After the introduction in 1994, by Okabe and Matsuda, of the notion of
semistar operation, many authors have investigated different aspects of this
general and powerful concept. A natural development of the recent work in this
area leads to investigate the concept of invertibility in the semistar setting.
In this paper, we will show the existence of a ``theoretical obstruction'' for
extending many results, proved for star-invertibility, to the semistar case.
For this reason, we will introduce two distinct notions of invertibility in the
semistar setting (called $\star$--invertibility and
quasi--$\star$--invertibility), we will discuss the motivations of these ``two
levels'' of invertibility and we will extend, accordingly, many classical
results proved for the $d$--, $v$--, $t$-- and $w$-- invertibility.
