  In this paper we count the number $N_n^{\text{tor}}(X)$ of $n$-dimensional
algebraic tori over $\mathbb{Q}$ whose Artin conductor of the associated
character is bounded by $X$. This can be understood as a generalization of
counting number fields of given degree by discriminant. We suggest a conjecture
on the asymptotics of $N_n^{\text{tor}}(X)$ and prove that this conjecture
follows from Malle's conjecture for tori over $\mathbb{Q}$. We also prove that
$N_2^{\text{tor}}(X) \ll_{\varepsilon} X^{1 + \varepsilon}$, and this upper
bound can be improved to $N_2^{\text{tor}}(X) \ll_{\varepsilon} X (\log X)^{1 +
\varepsilon}$ under the assumption of the Cohen-Lenstra heuristics for $p=3$.
