Special Value Formulas for Rankin-Selberg $p$-adic $L$-Functions

We discuss special value formulas for a $p$-adic $L$-function $L_p(f,\chi )$, where $f$ is a fixed newform and
$\chi$ varies over the space of Hecke characters of a fixed imaginary quadratic field, as well as some recent
applications. These formulas, first found by Bertolini, Darmon and Prasanna, relate $L_p(f,1)$, a value which is outside the range of interpolation defining $L_p$, to arithmetic invariants of cycles on varieties fibered over modular curves.