Dark solitons are coherently propagating nonlinear waves that are characterised by the localised absence of a background medium, which could be light (dark spot) or a matter-wave field rarefaction pulse). Here, we show that families of dark solitons characterised by their propagation velocity exist in superfluid Fermi gases. The energy-velocity dispersion and number of depleted particles completely determines the dynamics of dark solitons on a slowly-varying background density. For the unitary Fermi gas we determine these relations from general scaling arguments and conservation of local particle number. We predict solitons in a trapped Fermi gas to oscillate sinusoidally at the trap frequency reduced by a factor of $1/\sqrt{3}$. Numerical integration of the time-dependent Bogoliubov-de Gennes equation determines spatial profiles and soliton dispersion relations across the BEC-BCS crossover and proves consistent with the scaling relations at unitarity.