  The properties of matter are significantly modified by strong magnetic
fields, $B>>2.35\times 10^9$ Gauss ($1 G =10^{-4} Tesla$), as are typically
found on the surfaces of neutron stars. In such strong magnetic fields, the
Coulomb force on an electron acts as a small perturbation compared to the
magnetic force. The strong field condition can also be mimicked in laboratory
semiconductors. Because of the strong magnetic confinement of electrons
perpendicular to the field, atoms attain a much greater binding energy compared
to the zero-field case, and various other bound states become possible,
including molecular chains and three-dimensional condensed matter. This article
reviews the electronic structure of atoms, molecules and bulk matter, as well
as the thermodynamic properties of dense plasma, in strong magnetic fields,
$10^9G << B < 10^{16}G$. The focus is on the basic physical pictures and
approximate scaling relations, although various theoretical approaches and
numerical results are also discussed. For the neutron star surface composed of
light elements such as hydrogen or helium, the outermost layer constitutes a
nondegenerate, partially ionized Coulomb plasma if $B<<10^{14}G$, and may be in
the form of a condensed liquid if the magnetic field is stronger (and
temperature $<10^6$ K). For the iron surface, the outermost layer of the
neutron star can be in a gaseous or a condensed phase depending on the cohesive
property of the iron condensate.
