Consider an abelian variety $A$ over a number field $K$. For a finite extension $L/K$, the cardinality of the group $A(L)_{tors}$ of torsion points in $A(L)$ can be bounded polynomially in terms of the degree $[L:K]$. We describe the smallest real number $\gamma_A$ such that for any finite extension $L/K$ and $\varepsilon>0$, we have $|A(L)_{tors}| \leq C [L:K]^{\gamma_A+\varepsilon}$, where the constant $C$ depends only on $A$ and $\varepsilon$. Assuming the Mumford-Tate conjecture for $A$, our constant $\gamma_A$ agrees with the conjectured optimal exponent of Hindry and Ratazzi. We will discuss what is known concerning the images of the $\ell$-adic Galois representations associated to $A$.