  The low-temperature behavior of the electron phase coherence time,
$\tau_{\phi}$, in mesoscopic metal wires has been a subject of controversy
recently. Whereas theory predicts that $\tau_{\phi}(T)$ in narrow wires should
increase as $T^{-2/3}$ as the temperature $T$ is lowered, many samples exhibit
a saturation of $\tau_{\phi}$ below about 1 K. We review here the experiments
we have performed recently to address this issue. In particular we emphasize
that in sufficiently pure Ag and Au samples we observe no saturation of
$\tau_{\phi}$ down to our base temperature of 40 mK. In addition, the measured
magnitude of $\tau_{\phi}$ is in excellent quantitative agreement with the
prediction of the perturbative theory of Altshuler, Aronov and Khmelnitskii. We
discuss possible explanations why saturation of $\tau_{\phi}$ is observed in
many other samples measured in our laboratory and elsewhere, and answer the
criticisms raised recently by Mohanty and Webb regarding our work.
