  The recent growing interest in terahertz (THz) and sub-THz science and
technology is due to its many important applications in physics, astronomy,
chemistry, biology, and medicine. We review the problem of linear and
non-linear THz and sub-THz Josephson plasma waves in layered superconductors
and their excitations produced by moving Josephson vortices. We start by
discussing the coupled sine-Gordon equations for the gauge-invariant phase
difference of the order parameter in the junctions, taking into account the
effect of breaking the charge neutrality, and deriving the spectrum of
Josephson plasma waves. We also review surface and waveguide Josephson plasma
waves. We review the propagation of weakly nonlinear Josephson plasma waves
below the plasma frequency, which is very unusual for plasma-like excitations.
In close analogy to nonlinear optics, these waves exhibit numerous remarkable
features, including a self-focusing effect, and the pumping of weaker waves by
a stronger one. We also present quantum effects in layered superconductors,
specifically, the problem of quantum tunnelling of fluxons through stacks of
Josephson junctions. We discuss the Cherenkov and transition radiations of the
Josephson plasma waves produced by moving Josephson vortices. We also discuss
the problem of coherent radiation (superradiance) of the THz waves by exciting
uniform Josephson oscillations. The effects reviewed here could be potentially
useful for sub-THz and THz emitters, filters, and detectors.
