  We derive the equations for the propagation of relativistic ionization fronts
in both static and moving gases. We focus on the supersonic R-type phase that
occurs right after a source turns on, and we compare the nonrelativistic and
relativistic solutions for several important cases. Relativistic corrections
can be significant up until the light-crossing time of the equilibrium
Stromgren sphere. For a static medium, we obtain exact analytical solutions and
apply them to the illustrative problems of an O star in a molecular cloud and a
starburst in a high-redshift cosmological halo. Relativistic corrections can be
important at early times when the H II regions are small, as well as at later
times, if a density gradient causes the I-front to accelerate. For the
cosmologically-expanding IGM, we derive an analytical solution in the case of a
steady source and a constant clumping factor. Here relativistic corrections are
significant for short-lived, highly-luminous sources like QSOs at the end of
reionization, but negligible for weaker or higher-redshift sources. Finally, we
numerically calculate the evolution of relativistic I-fronts in the presence of
small-scale structure and infall, for a large galaxy undergoing a starburst and
a luminous, high-redshift QSO. For such strong and short-lived sources, the
relativistic corrections are quite significant, and small-scale structure can
decrease the size of the H II region by up to an additional ~25%. (abridged)
