An　adaptive　multigrid　method　for　semilinear　elliptic　equations　based　on　adaptive　multigrid　methods　and　on　multilevel　correction　methods　is　developed.　The　solution　of　a　semilinear　problem　is　reduced　to　a　series　of　linearised　elliptic　equations　on　the　sequence　of　adaptive　finite　element　spaces　and　semilinear　elliptic　problems　on　a　very　low　dimensional　space.　The　corresponding　linear　elliptic　equations　are　solved　by　an　adaptive　multigrid　method.　The　convergence　and　optimal　complexity　of　the　algorithm　is　proved　and　illustrating　numerical　examples　are　provided.　The　method　requires　only　the　Lipschitz　continuity　of　the　nonlinear　term.　This　approach　can　be　extended　to　other　nonlinear　problems,　including　Navier-Stokes　problems　and　phase　field　models.