We　consider　testing　the　mutual　independency　for　high-dimensional　data.　It　is　known　that　L-2-type　statistics　have　lower　power　under　sparse　alternatives　and　L-infinity-type　statistics　have　lower　power　under　dense　alternatives　in　high　dimensions.　In　this　paper,　we　develop　an　adaptive　test　based　on　Kendall＇s　tau　to　compromise　both　situations　of　the　alternative,　which　can　automatically　be　adapted　to　the　underlying　data.　An　adaptive　test　is　very　useful　in　practice　as　the　sparsity　or　density　for　a　data　set　is　usually　unknown.　In　addition,　we　establish　the　asymptotic　joint　distribution　of　L-2-type　and　L-infinity-type　statistics　based　on　Kendall＇s　tau　under　mild　assumptions　and　the　asymptotic　null　distribution　of　the　proposed　statistic.　Simulation　studies　show　that　our　adaptive　test　performs　well　in　either　dense　or　sparse　cases.　To　illustrate　the　usefulness　and　effectiveness　of　the　proposed　test,　real　data　sets　are　also　analysed.