Magnetic skyrmions have great potential for developing novel spintronic devices. The electrical manipulation of skyrmions has mainly relied on current-induced spin-orbit torques. A recent theoretical model suggested that the skyrmions could be more efficiently manipulated by surface acoustic waves (SAW), an elastic wave that can couple with magnetic moment through magnetoelastic effect. However, the directional motion of skyrmions that is driven by SAW is still missing. Here, we experimentally demonstrate the motion of Néel-type skyrmions in Ta/CoFeB/MgO/Tamultilayers driven by propagating SAW pulses from on-chip piezoelectric transducers. Our results revealthat the elastic wave with longitudinal and shear vertical displacements (Rayleigh wave) traps skyrmions, while the shear horizontal wave effectively drives the motion of skyrmions. In particular, a longitudinal motion along the SAW propagation direction and a transverse motion due to topological charge, are observed and further confirmed by our micromagnetic simulations. This work demonstrates a promising approach based on acoustic waves for manipulating skyrmions, which could offer new opportunities for ultra-low power spintronics.