This　paper　mainly　discusses　the　pth　moment　asymptotic　stability　and　the　exponential　stability　of　nonlinear　stochastic　functional　differential　equations　(SFDEs)　satisfying　the　local　Lipschitz　condition　but　not　the　linear　growth　condition.　These　new　conditions　assume　that　the　coefficients　of　SFDEs　are　polynomials　or　dominated　by　polynomials.　We　establish　some　sufficient　conditions　for　the　pth　moment　asymptotic　stability　and　the　exponential　stability　of　nonlinear　SFDEs　by　applying　some　novel　techniques.　Some　nontrivial　examples　are　provided　to　illustrate　our　results.