  Both major galaxies in the Local Group (LG) are surrounded by thin planes of
mostly co-orbiting satellite galaxies, the vast polar structure (VPOS) around
the Milky Way (MW) and the Great Plane of Andromeda (GPoA) around M31. We
summarize the current knowledge concerning these structures and compare their
relative orientations and properties in a common coordinate system. The
existence of coherent satellite structures motivates an investigation of the
distribution of the more distant non-satellite galaxies in the LG. This results
in the discovery of two planes (diameters of 1-2 Mpc) which contain almost all
nearby non-satellite galaxies. The two LG planes are surprisingly symmetric,
inclined by only 20 deg relative to the galactic disc of M31, similarly thin
(heights of ~60 kpc) and have near-to-identical offsets from the MW and M31.
They are inclined relative to each other by 35 deg. Comparing the plane
orientations with each other and with additional features indicates an intimate
connection between the VPOS and the GPoA. They are both polar to the MW, have
similar orbital directions and are inclined by ~45 deg relative to each other.
The Magellanic Stream approximately aligns with the VPOS and the GPoA, but also
shares its projected position and line-of-sight velocity trend with the
dominating structure of non-satellite dwarf galaxies. The proper motion of M31
indicates a prograde orbit of the MW-M31 system, the VPOS and the GPoA. The
alignment with other features such as the Supergalactic Plane and the
over-density in hypervelocity stars are discussed as well. We end with a
summary of proposed scenarios trying to explain the LG galaxy structures as
either originating from cosmological structures or from tidal debris of a past
galaxy encounter. There currently exists no full detailed model which
satisfactorily explains the existence of the thin symmetric LG planes.
(abridged)
