  We construct an anticyclotomic Euler system for the Rankin-Selberg
convolution of two modular forms, using $p$-adic families of generalized
Gross-Kudla-Schoen diagonal cycles. As applications of this construction, we
prove new cases of the Bloch-Kato conjecture in analytic ranks zero and one,
and a divisibility towards an Iwasawa main conjecture.
