  This is a survey of recent work on values of Rankin-Selberg $L$-functions of
pairs of cohomological automorphic representations that are {\it critical} in
Deligne's sense. The base field is assumed to be a CM field. Deligne's
conjecture is stated in the language of motives over $\QQ$, and express the
critical values, up to rational factors, as determinants of certain periods of
algebraic differentials on a projective algebraic variety over homology
classes. The results that can be proved by automorphic methods express certain
critical values as (twisted) period integrals of automorphic forms. Using
Langlands functoriality between cohomological automorphic representations of
unitary groups, which can be identified with the de Rham cohomology of Shimura
varieties, and cohomological automorphic representations of $GL(n)$, the
automorphic periods can be interpreted as motivic periods. We report on recent
results of the two authors, of the first-named author with Grobner, and of
Guerberoff.
