  We study in this paper the cluster category C(S,M) of a marked surface (S,M).
We explicitly describe the objects in C(S,M) as direct sums of homotopy classes
of curves in (S,M) and one-parameter families related to closed curves in
(S,M). Moreover, we describe the Auslander-Reiten structure of the category
C(S,M) in geometric terms and show that the objects without self-extensions in
C(S,M) correspond to curves in (S,M) without self-intersections. As a
consequence, we establish that every rigid indecomposable object is reachable
from an initial triangulation.

Representation Theory