Arithmetic Toric Varieties Toric varieties are fundamental objects of combinatorial algebraic geometry. The reason for this is that they are canonically associated to a fan in a lattice and may be viewed either as varieties or as objects in geometric combinatorics. They area also characterized as normal varieties that have an action of a diagonal (split) torus with a dense orbit. An arithmetic toric variety is a normal variety over a field k equipped with the action of a (non necessarily split) torus having a dense orbit. Extending scalars to the algebraic closure, they become a usual toric variety. Their classification involves is via nonabelian (Galois cohomology) which mixes the combinatorics with the symmetry of a discrete group action. In this talk, which represents joint work with Javier Elizondo, Paulo Lima-Filho, and Zach Teitler, I will introduce you to the topics of toric varieties and Galois cohomology, stating our classification theorem, present some examples, and discuss future work in this area. It is the foundation for our work with Clarence Wilkerson on equivariant cohomology for real toric varieties, which is the subject of a future talk.