In the third and the last volume of this series the authors concentrate on the study of arbitrary representation-infinite tilted algebras of Euclidean type, and give a reasonable description of their indecomposable modules, their module categories and the Auslander-Reiten quivers. [For Vol. 2 cf. Lond. Math. Soc. Student Texts 71 (2007;Zbl 1129.16001).]
The volume consists of six chapters, numbered from Chapter XV to Chapter XX. In Chapter XV the authors describe several classes of extensions and coextensions of algebras, in particular, one-point (co)extensions, tubular (co)extensions and branch (co)extensions. Tubular (co)extensions of concealed algebras of Euclidean type are described in details.
In Chapter XVI the authors study branches and finite line extensions and describe tilted algebras of an equioriented type \(\mathbb{A}_m\). Chapter XVII is devoted to the study of tilted algebras of Euclidean type, in particular, the authors present a complete description of arbitrary representation-infinite tilted algebras of Euclidean type and their module categories, due to Ringel.
In Chapter XVIII the authors turn their attention to the representation theory of wild hereditary algebras associated with acyclic quivers. In particular, they describe the shape of components in the regular part of the Auslander-Reiten quiver for such algebras. In Chapter XIX the authors introduce the concepts of tame and wild representation type and prove that concealed algebras of Euclidean type are tame and that concealed algebras of wild type are wild.
In the final Chapter XX the authors present (without proofs) selected results of the representation theory of finite-dimensional algebras, which are related to the material discussed in the previous chapters, and which are supposed to give the reader a feeling for prospectives of the theory.

Reviewer: Volodymyr Mazorchuk (Uppsala)