Anapole moments are parity-odd, time-reversal-even moments of the E1 projection of the electromagnetic current. Although it was recognized, soon after the discovery of parity violation in the weak interaction, that elementary particles and composite systems like nuclei must have anapole moments, it proved difficult to isolate this weak radiative correction. The first successful measurement, an extraction of the nuclear anapole moment of 133 Cs from the hyperfine dependence of the atomic parity violation, was obtained only recently. An important anapole moment bound in Tl also exists. We discuss these measurements and their significance as tests of the hadronic weak interaction, focusing on the mechanisms that operate within the nucleus to generate the anapole moment. The atomic results place new constraints on weak meson-nucleon couplings, ones we compare to existing bounds from a variety of p − p and nuclear tests of parity nonconservation.