  Nuclear anapole moments are parity-odd, time-reversal-even E1 moments of the
electromagnetic current operator. Although the existence of this moment was
recognized theoretically soon after the discovery of parity nonconservation
(PNC), its experimental isolation was achieved only recently, when a new level
of precision was reached in a measurement of the hyperfine dependence of atomic
PNC in 133Cs. An important anapole moment bound in 205Tl also exists. In this
paper, we present the details of the first calculation of these anapole moments
in the framework commonly used in other studies of hadronic PNC, a meson
exchange potential that includes long-range pion exchange and enough degrees of
freedom to describe the five independent $S-P$ amplitudes induced by
short-range interactions. The resulting contributions of pi-, rho-, and
omega-exchange to the single-nucleon anapole moment, to parity admixtures in
the nuclear ground state, and to PNC exchange currents are evaluated, using
configuration-mixed shell-model wave functions. The experimental anapole moment
constraints on the PNC meson-nucleon coupling constants are derived and
compared with those from other tests of the hadronic weak interaction. While
the bounds obtained from the anapole moment results are consistent with the
broad ``reasonable ranges'' defined by theory, they are not in good agreement
with the constraints from the other experiments. We explore possible
explanations for the discrepancy and comment on the potential importance of new
experiments.
