A survey of splitting theorems for abstract groups and their applications. Topics covered include preliminaries, early results, Bass-Serre theory, the structure of G-trees, Serre’s applications to SL₂ and length functions. Stallings’ theorem, results about accessibility and bounds for splittability. Duality groups and pairs; results of Eckmann and collaborators on PD² groups. Relative ends, the JSJ theorems and the splitting results of Kropholler and Roller on PDⁿ groups.

Notions of quasi-isometry, of hyperbolic group, and of its boundary. We recall that convergence groups on the circle are Fuchsian, and survey results relating properties of the action of a hyperbolic group on its boundary to the structure of the group. Types of isometric action of a group on a -tree, and the -tree of a valued field, with mention of the applications made by Culler, Shalen and Morgan. Rips’ theorem, and some of its applications.

Splittings over 2-ended groups and work of Sela and Bowditch, more general splitting theorems, characterisations of groups by their coarse geometry. Finally we survey the extent to which it is possible to push through the Thurston programme for PD³ complexes and pairs: despite many advances, there remain more conjectures than theorems.

Key words: accessibility, acylindrical, adapted splitting, almost invariant set, amalgamated free product, atoroidal, Bass-Serre theory, coarse geometry, convergence group, elementary deformation, end, Fuchsian group, graph of groups, group pair, HNN group, hyperbolic group, JSJ splitting, -tree, length function, nested sets, Poincaré duality group, quasi-isometry, splitting, track, vertex pair.
2000 Mathematics Subject Classification: 20E08, 20F32, 20G25, 20J05, 57M07, 57P10.