A survey of recent developments in the study of the existence of solutions of nonlinear variational equations and inequalities for quasilinear elliptic operators of the form \[ Au(x)=\sum_{| \alpha | \leq m}(-1)^{| \alpha |}D^{\alpha} A_{\alpha}(x,u(x),...,D^ mu(x)),\quad x\in \Omega \subset R^ N \] is given. We also sketch a Galerkin method which is useful for very strongly nonlinear operators, i.e. even the top order coefficients \(A_{\alpha}\) do not obey any polynomial growth condition.