  (Edited for length) The Zeeman effect is the only observational technique
available to measure directly the strength of magnetic fields in regions of
star formation. We review the physics of the Zeeman effect and its practical
use in both extended gas and in masers. We discuss observational results for
the five species for which the Zeeman effect has been detected in the ISM --
H~I, OH, and CN in extended gas and OH, CH$_3$OH, and H$_2$O in masers. These
species span densities from $\sim10$ cm$^{-3}$ to $\sim10^{10}$ cm $^{-3}$,
which allows magnetic fields to be measured over the full range of cloud
densities. However, there are significant limitations, including that only the
line-of-sight component of the magnetic field strength can usually be measured
and that there are often significant uncertainties about the physical
conditions being sampled, particularly for masers. We discuss statistical
methods to partially overcome these limitations. The results of Zeeman
observations are that the mass to magnetic flux ratio is subcritical (gravity
dominates magnetic support) at lower densities but supercritical for $N_H
\gtrsim 10^{22}$ cm$^{-2}$. Above $n_H\sim 300$ cm$^{-3}$, which is roughly the
density at which clouds typically become self-gravitating, the strength of
magnetic fields increases approximately as $B \propto n^{2/3}$, which suggest
that magnetic fields do not provide significant support at high densities. This
is consistent with high-density clouds being supercritical. However, magnetic
fields have a large range in strengths at any given density, so the role of
magnetic fields should differ significantly from one cloud to another. And for
maser regions the dependence of field strength on density may have a slightly
lower slope. Turbulent reconnection theory seems to best match the Zeeman
observational results.
