An ion linac is designed for acceleration of a single particle, which remains in synchronism with the accelerating fields and is called the synchronous particle. For an acceptable output beam intensity, restoring forces must be present so that those particles near the synchronous particle will have stable trajectories. Longitudinal restoring forces are produced when the beam is accelerated by an electric field that is increasing with time, and these forces produce phase and energy oscillations about the synchronous particle. The final energy of an ion that undergoes phase oscillations is approximately determined not by the field but by the geometry of the structure, which is tailored to produce a specific final synchronous energy. An exception is an ion linac built from an array of short independent cavities, each one capable of operating over a wide velocity range. In this case the final energy depends on the field and the phasing of the cavities. The longitudinal dynamics are different for relativistic electron linacs. After beam injection into electron linacs, the velocities approach the speed of light so rapidly that hardly any phase oscillations take place. The electrons initially slip relative to the wave to rapidly approach a final phase, which is maintained all the way to high energy. The final energy of each electron with a fixed phase depends on the accelerating field and on the value of the phase. In this chapter we develop the general equations of longitudinal dynamics in a linac, including the Liouvillian gap transformations used in computer simulation codes. This will be followed by special treatments of the dynamics in coupled-cavity ion linacs, independent cavity ion linacs, and low-energy electron beams injected into conventional electron-linac structures.