Robust　stability　has　been　one　of　the　most　popular　topics　in　the　study　of　stochastic　functional　differential　equations　with　Markovian　switching　(SFDEwMSs),　including　stochastic　delay　differential　equations　with　Markovian　switching　(hybrid　SDDEs).　Most　of　the　existing　results　on　the　robust　stability　require　that　the　drift　coefficient　f　and　diffusion　coefficient　g　of　the　stochastic　system　are　either　linear　or　nonlinear　with　linear　growth　condition.　Recently　Hu,　Mao　and　Zhang　(IEEE　Trans.　Autom.　Control　58:　2319-2332,　2013)　obtained　some　new　results　for　the　robust　stability　of　nonlinear　hybrid　SDDEs,　requiring　x(T)f　and　vertical　bar　g　vertical　bar(2)　to　be　bounded　by　polynomials　with　the　same　orders.　However,　there　are　many　SFDEwMSs　which　do　not　satisfy　the　above　requirement.　Hence　the　existing　criteria　on　the　robust　stability　are　not　applicable　and　we　see　the　necessity　to　develop　some　new　criteria.　Our　aim　in　this　paper　is　to　establish　some　new　criteria　for　the　robust　stability　of　a　class　of　SFDEwMSs,　where　x(T)f　and　vertical　bar　g　vertical　bar(2)　are　controlled　by　polynomials　with　different　orders.