Review of Zeeman Effect Observations of Regions of Star Formation

(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.