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Undoped and strongly photoexcited semiconductor superlattices with field-dependent recombination behave as excitable or oscillatory media with spatially discrete nonlinear convection and diffusion. Infinitely long, dc-current-biased superlattices behaving as excitable media exhibit wave fronts with increasing or decreasing profiles, whose velocities can be calculated by means of asymptotic methods. These superlattices can also support pulses of the electric field. Pulses moving downstream with the flux of electrons can be constructed from their component wave fronts, whereas pulses advancing upstream do so slowly and experience saltatory motion: they change slowly in long intervals of time separated by fast transitions during which the pulses jump to the previous superlattice period. Photoexcited superlattices can also behave as oscillatory media and exhibit wave trains.