This paper calculates the Grothendieck ring of the category \(\mathcal U\) of unstable modules over the Steenrod algebra, and of the quotient category \({\mathcal U}/{\mathcal N}il\) of \(\mathcal U\) by the subcategory of nilpotent modules. The results show that, for \(\mathcal U\), the Poincaré series determines the ring, and that a suitable variant of the Poincaré series does so for the quotient category. More precisely, the author shows that if two elements of the Grothendieck ring have the same Poincaré series then they must be equal. And he establishes which power series can occur as Poincaré series, thus identifying the Grothendieck ring as a subring of the power series ring. For the quotient category, one uses a ‘stabilised’ Poincaré series, where the coefficient of \(x^n\) is not the dimension of the module in degree \(n\), but rather the dimension in degree \(2^q n\) for \(q\) large enough - one may assume the module is reduced in which case this dimension is independent of \(q\) for sufficiently large \(q\). The paper works over the prime \(2\), but it is said that the results can be extended to all finite fields: “This amusing exercise is left to the reader.”