This paper studies Auslander-Reiten triangles in subcategories of triangulated categories. The main theorem shows that the Auslander-Reiten triangles in a subcategory are closely connected with the approximation properties of the subcategory. Namely, let C be an object in the subcategory C of the triangulated category T, and let
be an Auslander-Reiten triangle in T. Then under suitable assumptions, there is an Auslander-Reiten triangle
in C if and only if there is a minimal right-C-approximation of the form
The theory is used to give a new proof of the existence of Auslander-Reiten sequences over finite dimensional algebras.
(Received May 2007)