Summary: We prove that a self-avoiding walk on \(\mathbb Z^d\) is sub-ballistic in any dimension \(d \geq 2\). That is, writing \(||u||\) for the Euclidean norm of \(u \in \mathbb Z^d\), and \(\operatorname{P} _{\mathrm {SAW}_n}\) for the uniform measure on self-avoiding walks \(\gamma : \{0, \dotsc, n\} \to \mathbb Z^d\) for which \(\gamma _{0} = 0\), we show that, for each \(v > 0\), there exists \(\varepsilon >0\) such that, for each \(n \in \mathbb N\), \( \operatorname{P} _{\mathrm {SAW}_n}(\max\{||\gamma_k||:0\leq k \leq n\} \geq vn)\leq e^{-\varepsilon n}\).