  In this paper we make a systematic study of certain motivic cohomology
classes ("Rankin-Eisenstein classes") attached to the Rankin--Selberg
convolution of two modular forms of weight $\ge 2$. The main result is the
computation of the $p$-adic syntomic regulators of these classes. As a
consequence we prove many cases of the Perrin-Riou conjecture for
Rankin--Selberg convolutions of cusp forms.

Number Theory