  Waves and oscillations have been observed in the Sun's atmosphere for over
half a century. While such phenomena have readily been observed across the
entire electromagnetic spectrum, spanning radio to gamma-ray sources, the
underlying role of waves in the supply of energy to the outermost extremities
of the Sun's corona has yet to be uncovered. Of particular interest is the
lower solar atmosphere, including the photosphere and chromosphere, since these
regions harbor the footpoints of powerful magnetic flux bundles that are able
to guide oscillatory motion upwards from the solar surface. As a result, many
of the current- and next-generation ground-based and space-borne observing
facilities are focusing their attention on these tenuous layers of the lower
solar atmosphere in an attempt to study, at the highest spatial and temporal
scales possible, the mechanisms responsible for the generation, propagation,
and ultimate dissipation of energetic wave phenomena. Here, we present a
two-fold review that is designed to overview both the wave analyses techniques
the solar physics community currently have at their disposal, as well as
highlight scientific advancements made over the last decade. Importantly, while
many ground-breaking studies will address and answer key problems in solar
physics, the cutting-edge nature of their investigations will naturally pose
yet more outstanding observational and/or theoretical questions that require
subsequent follow-up work. This is not only to be expected, but should be
embraced as a reminder of the era of rapid discovery we currently find
ourselves in. We will highlight these open questions and suggest ways in which
the solar physics community can address these in the years and decades to come.
