Abstract: The talk begins with a brief introduction to the case of complexity zero. This is the theory of toric varieties, i.e., the equivariant torus embeddings. The complete description of toric varieties in terms of combinatorial data has made them to an important testing and example class in algebraic geometry. The aim of the talk is to present a related approach to the more general case of complexity one; here the biggest torus orbits are of codimension one. An important tool is the so-called Cox ring of the underlying variety. Applications to the resolution of singularities and the classification of Fano varieties will be discussed.