Abstract: The idea of studying rational maps by looking at the syzygies of the base ideal is a relatively new idea that has now become an important research topic. In this talk, we will discuss some recent results that lead to birationality criteria and formulas for the degree of rational maps that depend on the algebraic properties of the syzygies of the base ideal. Mainly, we will introduce a new algebra called "saturated special fiber ring" and we will discuss its relations with the degree and birationality of rational maps between irreducible projective varieties. Time permitting, we will also discuss some results in the problem of specializing the coefficients of a rational map. This talk is based on joint works with Laurent Busé and Carlos D’Andrea and with Aron Simis.