  We consider a normal complete rational variety with a torus action of
complexity one. In the main results, we determine the roots of the automorphism
group and give an explicit description of the root system of its semisimple
part. The results are applied to the study of almost homogeneous varieties. For
example, we describe all almost homogeneous (possibly singular) del Pezzo
k*-surfaces of Picard number one and all almost homogeneous (possibly singular)
Fano threefolds of Picard number one having a reductive automorphism group with
two-dimensional maximal torus.
