We study the cover time of multiple random walks on undirected graphs G = ( V , E ) . We consider k parallel, independent random walks that start from the same vertex. The speed-up is defined as the ratio of the cover time of a single random walk to the cover time of these k random walks. Recently, Alon et al. (2008) [5] derived several upper bounds on the cover time, which imply a speed-up of Ω ( k ) for several graphs; however, for many of them, k has to be bounded by O ( log n ) . They also conjectured that, for any 1 ⩽ k ⩽ n , the speed-up is at most O ( k ) on any graph. We prove the following main results: • We present a new lower bound on the speed-up that depends on the mixing time. It gives a speed-up of Ω ( k ) on many graphs, even if k is as large as n . • We prove that the speed-up is O ( k log n ) on any graph. For a large class of graphs we can also improve this bound to O ( k ) , matching the conjecture of Alon et al. • We determine the order of the speed-up for any value of 1 ⩽ k ⩽ n on hypercubes, random graphs and degree restricted expanders. For d -dimensional tori with d > 2 , our bounds are tight up to logarithmic factors. • Our findings also reveal a surprisingly sharp threshold behaviour for certain graphs, e.g., the d -dimensional torus with d > 2 and hypercubes: there is a value T such that the speed-up is approximately min { T , k } for any 1 ⩽ k ⩽ n .