Asymmetric light transport has significantly contributed to fundamental science and revolutionized advanced technology in various aspects such as unidirectional photonic devices, optical diodes, and isolators. While metasurfaces mold wave fronts at will with an ultrathin flat optical element, asymmetric transport of light cannot be fundamentally achieved by any linear system including linear metasurfaces. We report asymmetric transport of free-space light at nonlinear metasurfaces upon transmission and reflection. Moreover, we theoretically derive the nonlinear generalized Snell's laws that were experimentally confirmed by the anomalous nonlinear refraction and reflection. The asymmetric transport at optically thin nonlinear interfaces is revealed by the concept of a reversed propagation path. Such an asymmetric transport at metasurfaces opens a new paradigm for free-space ultrathin lightweight optical devices with one-way operation including unrivaled optical valves and diodes.