  Using a Green's function formulation of the superfluid current j_s, where a
momentum q_s is applied to the Cooper pair, we have calculated j_s as a
function of q_s, temperature, and impurity scattering for a two-band
superconductor. We consider both renormalized BCS and full strong-coupling
Eliashberg theory. There are two peaks in the current as a function of q_s due
to the two energy scales for the gaps and this can give rise to non-standard
behavior for the critical current. The critical current j_c, which is given as
the maximum in j_s, can exhibit a kink as a function of temperature as the
maximum is transferred from one peak to other. Other temperature variations are
also possible and the universal BCS behavior is violated. The details depend on
the material parameters of the system, such as the amount of coupling between
the bands, the gap anisotropy, the Fermi velocities, and the density of states
of each band. The Ginzburg-Landau relation between j_c, the penetration depth
lambda_L and thermodynamic critical field H_c, is modified. Using Eliashberg
theory with the electron-phonon spectral densities given from bandstructure
calculations, we have applied our calculations for j_s and j_c to the case of
MgB2 and find agreement with experiment.
