An algebraic variety X is called homogeneous if the automorphism group Aut(X) acts on X transitively. Examples of homogeneous varieties are homogeneous spaces G/H of an algebraic group G. In fact, the class of homogeneous varieties is much wider. Our aim is to describe homogeneous toric varieties. We say that a non-degenerate toric vatiety X is S-homogeneous if the subgroup of the group Aut(X) generated by root subgroups acts on X transitively. We obtain an explicit conbinatorial description of S-homogeneous toric varieties based on a lattice version of the linear Gale transform. The construction leads to a remarkable class of regular fans. We show that any non-degenerate homogeneous toric variety is a big open toric subset of a maximal S-homogeneous toric variety. In particular, every homogeneous toric variety is quasiprojective. We conjecture that any non-degenerate homogeneous toric variety is S-homogeneous.