Frequent assertions are made in the scientific literature that the magnetic field of light produces negligible effects and no magnetic dispersion at optical frequencies. In this talk however we present recent experimental results that reveal unexpectedly intense optical magnetic effects in non-metallic materials at moderate intensities, under non-resonant conditions that require a re-interpretation of the multipole expansion. We achieve quantitative agreement with the new results and examine prospects for practical realization of optical magnetic interactions at low intensities using resonant enhancement as well as the attainment of negative refractive indices in homogeneous transparent dielectrics.