Abstract: There is a general feeling that the difference between algebraic and topological thinking may be quite well expressed by comparing the features of abelian categories with those of triangulated categories. For a long time, abelian categories and triangulated categories were thought of as lying far apart, but the introduction of the cluster categories has narrowed the gap. For suitable abelian categories, there do exist finitely generated ideals such that the factor category is triangulated (the classical example being the module category of a self-injective algebra and the ideal generated by the indecomposable projective modules), conversely, as one now knows from the cluster categories, there may exist finitely generated ideals in a triangulated category such that the factor category is abelian. The lecture will discuss some some further examples where triangulated and abelian categories are obtained from each other is such a way. Parts of the lecture will rely on joint work with Pu Zhang (SJTU) which deals with the Gorenstein-projective representations of a quiver over the dual numbers. In addition I will report on recent considerations of Fan Kong (SJTU).