CONTENTS Introduction..............................................................................................5 1. General torus embeddings...................................................................7 1.1. Sets of subrings..............................................................................7 1.2. Complex of cones and torus embeddings. Basic properties and notation...............8 1.3. Jets of 1-p.s. at 0...........................................................................12 1.4. An application of torus embeddings. Desigularization of plane cusps by blowings up of the plane...............14 1.5. Some $G_m$-actions on torus embedding...................................18 2. Complex torus embeddings. Real and lion-negative parts..................20 2.1. Introduction...................................................................................20 2.2. The real non-negative part of the variety $X_Σ$...........................21 2.3. Bijection of $X_σ^{≥0}$ onto σ̆......................................................29 2.4. Real part of $X_Σ$. Reflexions......................................................35 3. Projective torus embeddings..............................................................37 3.1. Polyhedra......................................................................................37 3.2. Morse function...............................................................................41 3.3. Filtrations, cycles of orbits and projectivity.....................................46 4. Homology............................................................................................50 4.1. Poincaré polynomial......................................................................50 4.2. Chow ring and l-adic cohomology..................................................51 4.3. Cohomology ring of $X_Σ(R)$ with coefficients in Z/2Z..................52 4.4. Orientation.....................................................................................55 4.5. The 2-dimensional case, homology with integral coefficients.........56 References.............................................................................................62 Index.......................................................................................................64