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A complete list is given of all finite trivalent arc-transitive connected graphs on up to 768 vertices, completing and extending the Foster census. Several previously undiscovered graphs appear, including one on 448 vertices which is the smallest arc-transitive trivalent graph having no automorphism of order 2 which reverses an arc. The graphs on the list are classified according to type (as described by Djokovic and Miller in terms of group amalgams), and were produced with the help of a parallel program which finds all normal subgroups of low index in a finitely-presented group. Further properties of each graph are also given: its girth, diameter, Hamiltonicity, and whether or not it is bipartite.