  We present a categorification of four mutation finite cluster algebras by the
cluster category of the category of coherent sheaves over a weighted projective
line of tubular weight type. Each of these cluster algebras which we call
tubular is associated to an elliptic root system. We show that via a cluster
character the cluster variables are in bijection with the positive real Schur
roots associated to the weighted projective line. In one of the four cases this
is achieved by the approach to cluster algebras of Fomin-Shapiro-Thurston using
a 2-sphere with 4 marked points whereas in the remaining cases it is done by
the approach of Geiss-Leclerc-Schroer using preprojective algebras.

Representation Theory