The article is based on five lectures given at ICTP in September 2002. The first part is devoted to arc spaces including Kolchin’s theorem (if \(X\) is an integral scheme, then the arc space \(\mathcal L(X)\) is irreducible) and the Nash problem (concerning the case if \(X\) is singular). It is followed by a chapter on additive invariants of algebraic varieties as the Euler characteristic and the Hodge polynomial. Stable birational invariants are treated, some explanations concerning Poonen’s theorem are given and a motivic zeta function of Hasse–Weil type is studied. Chapter 3 is about motivic integration followed by a chapter with applications (arc invariants, Euler characteristics and modifications, birational Calabi–Yau varieties, stringy invariants). The last part deals with the McKay correspondence.
For the entire collection see [Zbl 1072.14002].