|In the first half of my talk I will begin with a classical theorem of Shimura on the critical values of the $L$-function of a holomorphic modular form. With this in the background I will then talk about Deligne's conjecture on the critical values of motivic $L$-functions while emphasizing the statement of this conjecture for symmetric power $L$-functions. In the second half of my talk I will describe some recent results of mine, partly in collaboration with Freydoon Shahidi, on the special values of Rankin-Selberg $L$-functions for $GL(n)\times GL(n-1)$. I will explain how we can use recent progress on Langlands functoriality to get results for the critical values of (odd) symmetric power $L$-functions. Time permitting, I hope to end my talk by sketching some work in progress, jointly with Günter Harder, on ratios of critical values.