Starting with the examples of Dirichlet's class number formula and the Birch--Swinnerton-Dyer conjecture, we give a broad overview of the expected relations between the arithmetic of algebraic objects (such as varieties) and the special values of their $L$-functions. Our formulation will deliberately tend toward lowbrow and down-to-earth, rather than aiming for the highest possible generality. \vskip .1in \noindent Note: Watkins is not a new faculty member at UCSD, but he is visiting one (Stein) for the week.