In　this　paper,　we　propose　a　novel　error　criterion　for　adaptive　filtering,　namely　the　smoothed　least　mean　p-power　(SLMP)　error　criterion,　which　aims　to　minimize　the　mean　p-power　of　the　error　plus　an　independent　and　scaled　smoothing　variable.　Some　important　properties　of　the　SLMP　criterion　are　presented.　In　particular,　we　show　that　if　the　smoothing　variable　is　symmetric　and　zero-mean,　and　p　is　an　even　number,　then　the　SLMP　error　criterion　will　become　a　weighted　sum　of　the　even-order　moments　of　the　error,　and　as　the　smoothing　factor　(i.e.　the　scale　factor)　is　large　enough,　this　new　criterion　will　be　approximately　equivalent　to　the　well-known　mean　square　error　(MSE)　criterion.　Based　on　the　proposed　error　criterion,　we　develop　a　new　adaptive　filtering　algorithm　and　its　kernelized　version,　and　derive　a　theoretical　value　of　the　steady-state　excess　mean　square　error　(EMSE).　Simulation　results　suggest　that　the　new　algorithms　with　proper　choice　of　the　smoothing　factor　may　perform　quite　well.　(C)　2015　Elsevier　Inc.　All　rights　reserved.