Recently　intensive　interest　has　been　raised　on　approximation　of　the　NP-hard　submodular　maximization　problem　due　to　their　theoretical　and　practical　significance.　In　this　work,　we　extend　this　line　of　research　by　focusing　on　the　simultaneous　approximation　of　multiple　submodular　function　maximization.　We　address　the　existence　and　nonexistence　results　for　both　deterministic　and　randomized　approximation　when　the　submodular　functions　are　symmetric　and　asymmetric,　respectively,　along　with　algorithmic　corollaries.　We　offer　complete　characterization　of　the　symmetric　case　and　partial　results　on　the　asymmetric　case.　©　2014,　Operations　Research　Society　of　China,　Periodicals　Agency　of　Shanghai　University,　and　Springer-Verlag　Berlin　Heidelberg.