Chapter 1 establishes the plausibility of “magnetic molecular clouds.” We show that the empirically known relations between spectral line width, density, and cloud size can be derived from a virial equilibrium model where gravity is balanced by the sum of magnetic and pressure support. And, we show that sub­stitution of measured density, cloud size, and line width measurements into the model can predict observed field strength (as compared with Zeeman observa­tions) to within a factor of two. In Chapter 2, we discuss the Zeeman effect and we present new measure­ ments of magnetic field strength based on OH and HI Zeeman observations at the Arecibo (305-m) and Green Bank (43-m) telescopes. The Barnard 1 (Bl) re­gion, in the Perseus Molecular Cloud Complex, is discussed in detail. We present OH spectral line intensity maps of the regions where the OH Zeeman effect was observed, which allow, for the first time, comparison of observed field strength values with predicted field strength values (based on observed line width, den­sity, and cloud size), using emission from a single molecular species. The fields measured are consistent with the equilibrium model of Chapter 1, and with other models. Chapter 3 is an investigation of the spatial structure of magnetic fields in molecular clouds. We present new optical polarization maps of dark clouds in Perseus, Taurus, and Ophiuchus. The polarization observed is attributed to pref­ erential extinction of background starlight by magnetically aligned dust grains in the clouds, and we analyze the polarization maps as maps of the projection of the magnetic field onto the plane of the sky. As in previous studies, the field traced by the polarization vectors is smooth in appearance, and has a well-defined uniform component, although it is not generally oriented at a special angle (i.e. 0° or 90°) to filamentary dark clouds. The dispersion in the position angle of the polarization vectors for an entire cloud complex is generally less than the varia­ tion in position angle of the long axis of filamentary clouds within the complex, suggesting that the magnetic field does not dominate the cloud structure on the size scale ~1 pc, although its energy, as discussed in Chapters 1 and 2, is still comparable to the energy in the gas motions.