| Resumen: A Fano variety is an algebraic variety with an ample anticanonical divisor - from the differential point of view, these are compact Kähler manifolds with positive definite Ricci curvature. Fano varieties play a central role in the general classification theory of algebraic varieties. After a short general introduction, we briefly recall the present state of the art of the classification of toric Fano varieties, i.e. those being almost homogeneous under the action of an algebraic torus. Then we consider more generally Fano varieties with a torus action of complexity one, i.e. the general torus orbit is of codimension one. We present combinatorial tools for the study of such varieties and discuss recent classification results in the case of threefolds with at most terminal singularities.