The　set　cover　problem　has　been　studied　extensively　for　many　years.　Submodular　function　plays　a　key　role　in　combinatorial　optimization.　Extending　the　set　cover　problem,　we　consider　three　submodular　cover　problems.　The　first　two　problems　minimize　linear　and　submodular　functions,　respectively,　subject　to　the　same　non-submodular　cover　constraint.　The　third　problem　minimizes　a　submodular　function　subject　to　non-submodular　cover　and　precedence　constraints.　Based　on　the　concepts　of　submodular　ratio　and　gap,　and　Lovasz　extension,　we　devise　greedy　and　primal-dual　approximation　algorithms　for　these　problems.