  We construct $p$-adic $L$-functions for Rankin--Selberg products of
automorphic forms of hermitian type in the anticyclotomic direction for both
root numbers. When the root number is $+1$, the construction relies on global
Bessel periods on definite unitary groups which, due to the recent advances on
the global Gan--Gross--Prasad conjecture, interpolate classical central
$L$-values. When the root number is $-1$, we construct an element in the
Iwasawa Selmer group using the diagonal cycle on the product of unitary Shimura
varieties, and conjecture that its $p$-adic height interpolates derivatives of
cyclotomic $p$-adic $L$-functions. We also propose the nonvanishing conjecture
and the main conjecture in both cases.
