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Abstract: A graph is symmetric if its automorphism group is transitive on the arc set of the graph. In this paper, we classify connected cubic symmetric graphs of order for each prime . All those symmetric graphs are explicitly constructed as normal Cayley graphs on some groups of order , and their automorphism groups are determined. There is a unique connected cubic symmetric graph of order . All connected cubic symmetric graphs of order for are regular covers of the three dimensional hypercube , and consist of four infinite families, of which two families exist if and only if and the other two families exist for each odd prime . In each family, there is a unique graph for a given order. [Copyright &y& Elsevier]