Modular symbols are cohomological interpretations of period integrals for cohomological automorphic representations. The Archimedean modular symbols, which are linear functionals on certain relative Lie algebra cohomology spaces, capture the Archimedean behavior of modular symbols. We define automorphic periods by investigating the rationality of these Archimedean modular symbols. This construction is an analogue of Deligne's periods for critical pure motives. By using Archimedean modular symbols and automorphic periods, together with the rationality of certain Eisenstein cohomology spaces and cuspidal cohomology spaces, we obtain some rationality results for critical values of standard L-functions and Rankin-Selberg L-functions. These results align with Blasius's conjecture. The talk is based on some recent works joint with Dihua Jiang, Yubo Jin, Jiang-Shu Li, Dongwen Liu, and Fangyang Tian.