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.