  Driven by increasingly complete observational knowledge of systems of
satellite galaxies, mutual spatial alignments and relations in velocities among
satellites belonging to a common host have become a productive field of
research. Numerous studies have investigated different types of such
phase-space correlations, and were met with varying degrees of attention by the
community. The Planes of Satellite Galaxies issue is maybe the best-known
example, with a rich field of research literature and an ongoing, controversial
debate on how much of a challenge it poses to the LCDM model of cosmology.
Another type of correlation, the apparent excess of close pairs of dwarf
galaxies, has received considerably less attention despite its reported tension
with LCDM expectations. With the expansion of proper motion measurements,
largely driven by Gaia, other peculiar phase-space correlations were uncovered
among the Milky Way satellites. Examples are the tangential velocity excess of
satellites compared to cosmological expectations, and the unexpected preference
of satellites to be close to pericenter. At the same time, other kinds of
correlations have been found to be more in line with cosmological expectations,
specifically lopsided satellite galaxy systems and the accretion of groups of
satellite galaxies. The latter has mostly been studied in cosmological
simulations thus far, but offers the potential to address some of the other
issues by providing a way to produce correlations among the orbits of a group's
satellite galaxy members. This review is the first to provide an introduction
to the highly active field of phase-space correlations among satellite galaxy
systems. It summarizes existing, recent research and highlights
interdependencies between the different, currently almost exclusively
individually considered types of correlations. Future prospects are also
briefly discussed. (abridged)
