  In this paper, we prove Deligne's conjecture on the algebraicity of critical
values of symmetric power $L$-functions associated to modular forms of weight
greater than four. We also prove new cases of Blasius' conjecture on the
algebraicity of critical values of tensor product $L$-functions associated to
modular forms, and an algebraicity result on critical values of Rankin-Selberg
$L$-functions for ${\rm GL}_n \times {\rm GL}_2$ in the unbalanced case, which
extends the previous results of Furusawa and Morimoto for ${\rm SO}(V) \times
{\rm GL}_2$. These are applications of our main result on the algebraicity of
ratios of special values of Rankin-Selberg $L$-functions.
