Starting with the "Leibniz" formula for π

π/4 = 1 - 1/3 + 1/5 - 1/7 + ...

the special values of Dirichlet L-functions have long been a source of fascination and frustration. From Euler's solution in 1734 of the Basel problem to Apery's proof in 1978 that zeta(3) is irrational, our progress on understanding the arithmetic of these numbers has been limited. In this talk, we discuss some results (old and new) about these numbers.