The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties over k. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. I will focus on Manin's conjecture and on its motivic analogue, which predicts the behavior of moduli spaces of curves of given degree on some algebraic varieties, and may be formulated in terms of the convergence properties of a series called the motivic height zeta function. This will lead me to explain how some power series with coefficients in the Grothendieck ring can be endowed with an Euler product decomposition.