  This is a detailed write-up of Koll\'ar's course at the EMS summer school in
Algebraic Geometry in Eger, Hungary, August 1996. The topics include
definitions and examples of rational and unirational varieties, with special
attention to varieties defined over non-algebraically closed fields, Segre's
theorem on non-rationality of cubic surfaces of Picard number one, Manin's
theorem that birationally equivalent cubic surfaces of Picard number one are
projectively equivalent, and Koll\'ar's method for constructing examples of low
degree non-rational hypersurfaces. Includes many exercises and their solutions.
