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Given a real fan in a real space consisting of real convex polyhedral cones, we construct a complete real fan which contains the fan, by two completely different methods. The first one is purely combinatorial and a proof of a related version was sketched earlier by Ewald. The second one is based on Nagata's method of imbedding an abstract variety into a complete variety. For the second method, we introduce the theory of Zariski-Riemann space of a fan.

Toric Fano varieties are algebraic varieties associated with a special class of convex polytopes in R". We extend results of V. E. Voskresenskij and A. A. Klyachko about the classification of such varieties using a purely combinatorial method of proof.