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.