To ascertain the absolute abundances of even the most common elements in astrophysical objects has proven to be remarkably difficult. Perhaps the most astonishing example is the Sun itself! The recent downward revision by up to 50% in solar abundances of light elements such as C, N, O and Ne, obtained from high-resolution spectroscopy and sophisticated 3-D radiative transfer models, has led to serious discrepancies with presumbly accurate Helioseismological observations and stellar models. However, the answer may lie elsewhere: the atomic physics of astrophysical opacities. Preliminary results reveal that large-scale atomic calculations may be needed with unprecedented accuracy in order to describe resonances in bound-free cross sections, heretofore treated as bound-bound lines. Another example is the large scatter in element abundance ratios obtained from otherwise well-studied objects, bright H II regions such as planetary nebulae. To wit: The Ne/S abundances ratios from extragalactic H II regions observed with SPITZER show variations of up to a factor of 3 relative to the canonical (!) solar values. Also problematic is the dichotomy in abundances (by up to an order of magnitude or more) derived from collisionally excited forbidden lines (CEL) and recombination lines (REL); for example, in oxygen abundances inferred from [O III] CEL as opposed to O II REL, both formed from the same ion. Again, recent work shows that the resolution of the problem may depend on highly detailed atomic physics: the effect of relativistic fine structure on resonances in very low energy photoionization cross sections.