These are lecture notes for the class I am teaching in Winter 2017.

Lecture 1. Definitions and first examples.

Lecture 2. Cubic hypersurfaces I: Rationality of certain cubic hypersurfaces.

Lecture 3. Cubic hypersurfaces II: Unirationality.

Lecture 4. Unirationality of general hypersurfaces of small degree.

Lecture 5. Rationality and unirationality for surfaces.

Lecture 6. The Artin-Mumford example.

Lecture 7. Stable rationality and decomposition of the diagonal.

Appendix 1. Review of singular cohomology.

Appendix 2. An introduction to etale cohomology and the Brauer group.

Appendix 3. An overview of Chow groups.