Driven by the 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 ΛCDM 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 ΛCDM expectations. With the fast expansion of proper motion measurements in recent years, largely driven by the Gaia mission, other peculiar phase-space correlations have been uncovered among the satellites of the Milky Way. Examples are the apparent tangential velocity excess of satellites compared to cosmological expectations, and the unexpected preference of satellites to be close to their pericenters. 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 it 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. The emphasis is on summarizing existing, recent research and highlighting interdependencies between the different, currently almost exclusively individually considered types of correlations. Future prospects in light of upcoming observational facilities and our ever-expanding knowledge of satellite galaxy systems beyond the Local Group are also briefly discussed.