Observations of magnetic fields in molecular clouds are essential for understanding their role in the evolution of dense clouds and in the star formation process. The two extreme-case models of star formation are the weak magnetic field scenario, in which turbulence drives molecular cloud evolution and therefore star formation, and the strong magnetic field scenario, with magnetic fields governing molecular cloud formation and ambipolar diffusion ultimately driving star formation. Continuum and spectral line polarization observations make it possible to infer magnetic field strengths and morphologies, and therefore to test which of the two star formation scenarios dominates. The observational techniques available for such observations - the Zeeman effect, mapping of polarized dust emission, and mapping of linearly polarized line emission - are briefly described, together with how these observations can test star formation theory. Examples of the existing observations and the current state of the testing of magnetically controlled star formation are presented. Finally, the prospects for the future of study of magnetic fields in molecular clouds and their role in the star formation process is discussed.