  In 1994, Matsuda and Okabe introduced the notion of semistar operation,
extending the "classical" concept of star operation. In this paper, we
introduce and study the notions of semistar linkedness and semistar flatness
which are natural generalizations, to the semistar setting, of their
corresponding "classical" concepts. As an application, among other results, we
obtain a semistar version of Davis' and Richman's overring-theoretical theorems
of characterization of Pr\"ufer domains for Pr\"ufer semistar multiplication
domains.
