It is predicted magnetic skyrmions can be controllably moved on surfaces using a focused laser beam. Here an absorbed power of the order 1 mW, focused to a spot-size of the order 10 μm, results in a local temperature increase of around 50 K, and a local perpendicular strain of the order 10−3 due to the thermo-elastic effect. For positive magneto-elastic coupling this generates a strong attractive force on skyrmions due to the magneto-elastic effect. The resultant motion is dependent on forces due to i) gradients in the local strain-induced magnetic anisotropy, ii) gradients in the effective anisotropy due to local temperature gradients, and magnetic parameters temperature dependences, and iii) Magnus effect acting on objects with non-zero topological number. Using dynamical magneto-thermo-elastic modelling, it is predicted skyrmions can be moved with significant velocities (up to 80 m/s shown), both for ferromagnetic and antiferromagnetic skyrmions, even in the presence of surface roughness. This mechanism of controllably moving single skyrmions in any direction, as well as addressing multiple skyrmions in a lattice, offers a new approach to constructing and studying skyrmionic devices with all-optical control.