  We study the effects of very strong magnetic fields on the equation of state
(EOS) in multicomponent, interacting matter by developing a covariant
description for the inclusion of the anomalous magnetic moments of nucleons.
For the description of neutron star matter, we employ a field-theoretical
approach which permits the study of several models which differ in their
behavior at high density. Effects of Landau quantization in ultra-strong
magnetic fields ($B>10^{14}$ Gauss) lead to a reduction in the electron
chemical potential and a substantial increase in the proton fraction. We find
the generic result for $B>10^{18}$ Gauss that the softening of the EOS caused
by Landau quantization is overwhelmed by stiffening due to the incorporation of
the anomalous magnetic moments of the nucleons. In addition, the neutrons
become completely spin polarized. The inclusion of ultra-strong magnetic fields
leads to a dramatic increase in the proton fraction, with consequences for the
direct Urca process and neutron star cooling. The magnetization of the matter
never appears to become very large, as the value of $|H/B|$ never deviates from
unity by more than a few percent. Our findings have implications for the
structure of neutron stars in the presence of large frozen-in magnetic fields.
