  We obtain exact traveling-wave solutions of the coupled nonlinear partial
differential equations that describe the dynamics of two classical scalar
fields in 1+1 dimensions. The solutions are kinks interpolating between
neighboring vacua. We compute the classical kink mass and show that it
saturates a Bogomol'nyi-type bound. We also present exact traveling-wave
solutions of a more general class of models. Examples include coupled $\phi^4$
and sine-Gordon models.
