Although the subject of this meeting is triggered star formation in a turbulent interstellar medium, it remains unsettled what role magnetic fields play in the star formation process. This paper briefly reviews star formation model predictions for the ratio of mass to magnetic flux, describes how Zeeman observations can test these predictions, describes new results – an extensive OH Zeeman survey of dark cloud cores with the Arecibo telescope, and discusses the implications. Conclusions are that the new data support and extend the conclusions based on the older observational results – that observational data on magnetic fields in molecular clouds are consistent with the strong magnetic field model of star formation. In addition, the observational data on magnetic field strengths in the interstellar medium strongly suggest that molecular clouds must form primarily by accumulation of matter along field lines. Finally, a future observational project is described that could definitively test the ambipolar diffusion model for the formation of cores and hence of stars.