  This Special Topic focuses on magnetohydrodynamic (MHD) processes in deep
interiors of planets, in which their fluid dynamos are in operation. The
dynamo-generated, global, magnetic fields provide a background for our
solar-terrestrial environment. Probing the processes within the dynamos is a
significant theoretical and computational challenge and any window into
interior dynamics greatly increases our understanding. Such a window is
provided by exploring rapid dynamics, particularly MHD waves about the
dynamo-defined basic state. This field is the subject of current attention as
geophysical observations and numerical modellings advance. We give here
particular attention to torsional Alfv\'{e}n waves/oscillations and magnetic
Rossby waves, which may be regarded as typical axisymmetric and nonaxisymmetric
modes, respectively, amongst a wide variety of wave classes of the
rapidly-rotating MHD fluids. The excitation of those waves is being evidenced
for the geodynamo, whilst also being suggested for Jupiter. We shall overview
their dynamics, summarise our current understanding, and give open questions
for future perspectives.
