This　paper　is　concerned　with　the　stability　of　the　traveling　front　solutions　with　critical　speeds　for　a　class　of　p-degree　Fisher-type　equations.　By　detailed　spectral　analysis　and　sub-supper　solution　method,　we　first　show　that　the　traveling　front　solutions　with　critical　speeds　are　globally　exponentially　stable　in　some　exponentially　weighted　spaces.　Furthermore　by　Evans＇s　function　method,　appropriate　space　decomposition　and　detailed　semigroup　decaying　estimates,　we　can　prove　that　the　waves　with　critical　speeds　are　locally　asymptotically　stable　in　some　polynomially　weighted　spaces,　which　verifies　some　asymptotic　phenomena　obtained　by　numerical　simulations.