We　consider　the　diffraction　of　an　electromagnetic　plane　wave　by　a　biperiodic　structure.　This　paper　is　concerned　with　a　numerical　solution　of　the　diffraction　grating　problem　for　three-dimensional　Maxwell＇s　equations.　Based　on　the　Dirichlet-to-Neumann　(DtN)　operator,　an　equivalent　boundary　value　problem　is　formulated　in　a　bounded　domain　by　using　a　transparent　boundary　condition.　An　a　posteriori　error　estimate-based　adaptive　edge　finite　element　method　is　developed　for　the　variational　problem　with　the　truncated　DtN　operator.　The　estimate　takes　account　of　both　the　finite　element　approximation　error　and　the　truncation　error　of　the　DtN　operator,　where　the　former　is　used　for　local　mesh　refinements　and　the　latter　is　shown　to　decay　exponentially　with　respect　to　the　truncation　parameter.　Numerical　experiments　are　presented　to　demonstrate　the　competitive　behaviour　of　the　proposed　method.