The　paper　is　concerned　with　the　three-dimensional　electromagnetic　scattering　from　a　large　open　rectangular　cavity　that　is　embedded　in　a　perfectly　electrically　conducting　infinite　ground　plane.　By　introducing　a　transparent　boundary　condition,　the　scattering　problem　is　formulated　into　a　boundary　value　problem　in　the　bounded　cavity.　Based　on　the　Fourier　expansions　of　the　electric　field,　the　Maxwell　equation　is　reduced　to　one-dimensional　ordinary　differential　equations　for　the　Fourier　coefficients.　A　fast　algorithm,　employing　the　fast　Fourier　transform　and　the　Gaussian　elimination,　is　developed　to　solve　the　resulting　linear　system　for　the　cavity　which　is　filled　with　either　a　homogeneous　or　a　layered　medium.　In　addition,　a　novel　scheme　is　designed　to　evaluate　rapidly　and　accurately　the　Fourier　transform　of　singular　integrals.　Numerical　experiments　are　presented　for　large　cavities　to　demonstrate　the　superior　performance　of　the　proposed　method.　(C)　2021　Elsevier　Inc.　All　rights　reserved.