Abstract. We review theoretical models of the early stages of star formation, in which gravitational collapse is strongly regulated by magnetic fields and the associated process of ambipolar diffusion. We discuss results of numerical simulations and analytical studies of core formation and collapse, which can be directly tested against observation. We also focus on recent data which are relevant to this theory of star formation, such as: observations of extended infall in protostellar cores, estimates of evolutionary timescales ∼ 1 Myr for cores, measured mass-to-flux ratios of cores, and the relative alignment of polarization vectors with apparent cloud elongation. It is shown that in all of these areas, the data remain compatible with magnetic collapse models which lie within the observationally allowable range of parameter space. Other areas of interest (protostellar accretion rates and the presence of core edges) and issues that remain unresolved or under study (the role of non-thermal motions and cluster formation) are also discussed. Moreover, we highlight some differences between our model predictions and those of highly turbulent star formation models, and discuss how these differences can be distinguished observationally. “Never multiply explanations or make them more complicated than necessary. An explanation should be as simple and direct as possible.” “No more things should be presumed than is necessary.