We will discuss our recent proof (joint work with C. Greither) of a conjecture linking $\ell$-adic realizations of $1$-motives and special values of equivariant $L$-functions in characteristic $p$, refining earlier results of Deligne and Tate. As a consequence, we will give proofs (in the characteristic $p$ setting) of various central classical conjectures on special values of $L$-functions, namely those due to Coates-Sinnott, Brumer-Stark, and Gross. Also, we will indicate how this theory can be extended to characteristic $0$.