A　novel　metric,　called　kernel-based　conditional　mean　dependence　(KCMD),　is　proposed　to　measure　and　test　the　departure　from　conditional　mean　independence　between　a　response　variable　Y　and　a　predictor　variable　X,　based　on　the　reproducing　kernel　embedding　and　the　Hilbert-Schmidt　norm　of　a　tensor　operator.　The　KCMD　has　several　appealing　merits.　It　equals　zero　if　and　only　if　the　conditional　mean　of　Y　given　X　is　independent　of　X,　i.e.　E(Y　vertical　bar　X)　=　E(Y)　almost　surely,　provided　that　the　employed　kernel　is　characteristic;　it　can　be　used　to　detect　all　kinds　of　conditional　mean　dependence　with　an　appropriate　choice　of　kernel;　it　has　a　simple　expectation　form　and　allows　an　unbiased　empirical　estimator.　A　class　of　test　statistics　based　on　the　estimated　KCMD　is　constructed,　and　a　wild　bootstrap　test　procedure　to　the　conditional　mean　independence　is　presented.　The　limit　distributions　of　the　test　statistics　and　the　bootstrapped　statistics　under　null　hypothesis,　fixed　alternative　hypothesis　and　local　alternative　hypothesis　are　given　respectively,　and　a　data-driven　procedure　to　choose　a　suitable　kernel　is　suggested.　Simulation　studies　indicate　that　the　tests　based　on　the　KCMD　have　close　powers　to　the　tests　based　on　martingale　difference　divergence　in　monotone　dependence,　but　excel　in　the　cases　of　nonlinear　relationships　or　the　moment　restriction　on　X　is　violated.　Two　real　data　examples　are　presented　for　the　illustration　of　the　proposed　method.　(C)　2021　Elsevier　B.V.　All　rights　reserved.