To investigate the cause of anomalously bright resonance lines of helium in the Sun, we have studied the magnitude of the enhancements and some proposals made to explain them. Calculations in new semiempirical solar models indicate that the resonance lines of helium are enhanced by factors of 2-5 for He i and between 2 and 9 for He ii, depending on the elemental abundances assumed. These enhancements are substantially less than earlier work has suggested, with the differences arising from radiative transfer not only in helium lines but also in lines of less abundant elements. Photon scattering, even with small line center optical depths (of order unity or less) throughout the transition region, is shown to significantly modify line intensities and their center-to-limb variation. This effect has important consequences both for our analysis and also for the analysis of solar EUV lines in general, including emission measure analyses. We have re-examined some proposals to explain the enhancements based on the ''ionizing plasma'' picture. The proposals include explicit heating (the ''burst'' picture) and heating via advection (diffusive and/or flowing models and the ''velocity redistribution'' [VR] proposal). We argue that the original VR mechanism must be modified to include kinetic effects for helium atoms, which reduce the effects of VR. The VR mechanism also naturally predicts helium lines that are blueshifted relative to lines less sensitive to VR, which contradicts observations for the He i 584 8 line. Ionizing plasma models also potentially lead to enhancement of Án ! 1 transitions (n is the principal quantum number) in other atoms and ions. Existing observational evidence for enhanced Án ! 1 transitions in other ions is weak. We conclude that the ionizing plasma scenarios are not the sole cause of the helium enhancement. Instead, in a companion paper, we propose that the thermodynamic properties of noble gases in the upper chromosphere will lead to enhancement of their spectral lines if cross-field diffusion into coronal plasma is important. Finally, we show that, surprisingly, multifluid flows of the kind computed by Fontenla and colleagues can reproduce observed intensities of several lines of C and Si and may contain a resolution to the helium enhancement problem.