Gen. Relativ. Gravit. (2006) 38(6): 1003–1015 DOI 10.1007/s10714-006-0283-4 RESEARCH ARTICLE G. F. R. Ellis The Bianchi models: Then and now Received: 12 December 2005 / Published online: 13 May 2006 C  Springer Science and Business Media Inc., New York 2006 1 The Bianchi universes The Bianchi universe models are spatially homogeneous cosmological models that in general are anisotropic. Such cosmologies provide interesting generalisations of the standard Friedmann-Lemaˆ ıtre models of cosmology (which are based on the spatially homogeneous and isotropic Robertson-Walker geometries, with spatial sections of constant curvature). The Bianchi models are defined to be the family of cosmologies in which there is a 3-dimensional group of isometries G 3 acting on spacelike 3-surfaces; making these surfaces of homogeneity in space-time (all physical quantities are necessarily constant on them). They are characterised in terms of their specific 3-dimensional symmetry groups, originally classified by L Bianchi [2, 3]. To be of cosmological interest, the space-times must be non- empty (T αβ = 0) with a preferred 4-velocity field u a determining the world lines of fundamental observers [17]. The simply transitive group of isometries G 3 is generated by Killing vectors ξ ν (ν = 1, 2, 3) with structure constants C γ αβ , defined by [ξ α ,ξ β ]= C γ αβ ξ γ , C γ αβ = C γ [αβ] (1) The group G 3 might be a subgroup of a larger multiply transitive symmetry group G 4 or G 6 , in which case in general there will be several different simply transitive subgroups G 3 . The focus in Bianchi models is on the existence of sim- ply transitive groups because in that case there are simple representations of the G. F. R. Ellis (B ) Department of Mathematics and Applied Mathematics, University of Cape Town, 7701 Rondebosch, South Africa; School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK E-mail: ellis@maths.uct.ac.za