  We construct motivic cohomology classes attached to Rankin--Selberg
convolutions of modular forms of weights $\ge 2$, show that these vary
analytically in p-adic families, and relate their image under the p-adic
regulator map to values of L-functions. As consequences, we prove new cases of
Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for
Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin
characters; and we prove one inclusion in the Iwasawa main conjecture for a
single modular form over an imaginary quadratic field.
