Toric varieties lie at the crossroads of many areas of mathematics, including algebraic geometry, algebraic topology, representation theory, mathematical physics, and algebraic combinatorics. Their explicit combinatorial descriptions make them a natural mathematical laboratory for testing general theories and exploring geometric phenomena.

This short course introduces the basic theory of toric varieties, with an emphasis on explicit computations and concrete examples. The main topics include:

1) The construction of toric varieties from cones and fans.

2) Algebraic torus actions on toric varieties and their orbit structures.

3) Explicit presentations of the ordinary and equivariant cohomology rings of projective toric manifolds.

4) Computations of intersection numbers and characteristic classes.

5) Nonnegative parts of toric varieties and their applications.