Group-theoretic orbit decidability

Abstract A recent collection of papers in the last years have given a renovated interest to the notion of orbit decidability. This is a new quite general algorithmic notion, connecting with several classical results, and closely related to the study of the conjugacy problem for extensions of groups. In the present survey we explain several of the classical results closely related to this concept, and we explain the main ideas behind the recent connection with the conjugacy problem made by Bogopolski–Martino–Ventura in [Trans. Amer. Math. Soc. 362 (2010), 2003–2036]. All the consequences up to date, published in several other papers by other authors, are also commented and reviewed.