We study random walks on the giant component of the Erdős-Rényi random graph ${\cal G}(n,p)$ where $p=λ/n$ for $λ>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $\log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(\log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $(ν{\bf d})^{-1}\log n \pm (\log n)^{1/2+o(1)}$, where $ν$ and ${\bf d}$ are the speed of random walk and dimension of harmonic measure on a ${\rm Poisson}(λ)$-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.