[For the entire collection see Zbl 0626.00024.]
The title of the paper practically says what this nicely written survey is about. The subtitles even further indicate what is presented in the paper and therefore we list them: § 1. Introduction; § 2. Geometric aspects of homology of groups; § 3. Connectedness at infinity of a locally finite complex; § 4. Connectedness for finitely presented groups; § 5. The (n-1)-shape of a group of type \({\mathcal F}(n)\); 6. The shape of a group of type \({\mathcal F}\); § 7. Pseudo-proper homotopy; § 8. The shape of a group of type \({\mathcal F}(\infty)\); § 9. The shape theory of groups. To clarify the symbols in the subtitles we point out that a group G is of type \({\mathcal F}(n)\) if the associated Eilenberg MacLane complex K(G,1) can be chosen to have finite n-skeleton, of type \({\mathcal F}(\infty)\) if K(G,1) can be chosen to have finite skeleta and of type \({\mathcal F}\) if K(G,1) can be chosen to be finite. The author proposes how to assign different shape theoretic notions such as (n-1)-shape of G, pro-(n-1)-type of G, shape of G etc. to a group G of a certain type. The crucial idea comes from the recognition that some shape theoretic properties of the universal cover of K(G,1) depend only on G. Known results related to this idea are organized and elaborated and also many open questions are discussed and posed.