Suppose that $\eta$ is a Schramm-Loewner evolution (SLE) in a smoothly bounded simply connected domain $D\subset C$ and that $\varphi$ is a conformal map from $D$ to a connected component of $D\setminus \eta ([0,t])$ for some $t>0$. The multifractal spectrum of $\eta$ is the function $(-1,1)\to [0,\infty )$ which, for each $s\in (-1,1)$, gives the Hausdorff dimension of the set of points $x\in \partial D$ such that $|\varphi \prime ((1-ϵ)x)|=ϵ-s+o(1)$ as $ϵ\to 0$. We rigorously compute the a.s. multifractal spectrum of SLE, confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE and we obtain a new derivation of the a.s. Hausdorff dimension of the SLE curve for $\kappa \le 4$. Our results also hold for the SLE processes with general vectors of weight $\rho ―$.