In　this　paper,　the　bases　of　a　semimodule　over　a　commutative　semiring　R　are　investigated.　Some　properties　and　characterizations　of　the　bases　are　discussed　and　some　equivalent　conditions　for　a　basis　to　be　a　free　basis　in　a　finitely　generated　free　semimodule　over　R　are　given.　The　different　possible　cardinalities　for　a　basis　in　a　finitely　generated　free　semimodule　over　R　are　considered　and　some　equivalent　descriptions　are　obtained　for　a　commutative　semiring　R　satisfying　the　property　that　any　two　bases　for　a　finitely　generated　free　R-semimodule　have　the　same　cardinality.　Partial　results　obtained　in　the　paper　develop　and,　generalize　the　corresponding　results　for　commutative　join-semirings　and　for　commutative　zerosumfree　semirings.　(C)　2013　Elsevier　Inc.　All　rights　reserved.