  Many mechanical systems have configuration spaces that admit symmetries.
Mathematically, such symmetries are modelled by the action of a group on a
topological space. Several variations of topological complexity have emerged
that take symmetry into account in various ways, either by asking that the
motion planners themselves admit compatible symmetries, or by exploiting the
symmetry to motion plan between functionally equivalent configurations. We will
survey the main definitions due to Colman-Grant, Lubawski-Marzantowicz,
B\l{}aszczyk-Kaluba and Dranishnikov, and some related notions. We conclude
with a short list of open problems.
