We report, to the best of our knowledge, the first experimental observation of vector surface solitons, which form at the edge and in the corner of two-dimensional laser-written waveguide arrays. These elliptically polarized vector states are composed of two orthogonally polarized components. They exist only above a power threshold and bifurcate from scalar surface solitons. The components of a vector soliton may have substantially different degrees of localization in certain parameter ranges.