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.