We design adaptive high-order Galerkin methods for the solution of linear elliptic problems and study their performance. We start by considering adaptive Fourier-Galerkin methods (ADFOUR), which offer unlimited approximation power only restricted by solution and data regularity. Next, we turn our attention to the hp-version of the finite element method, for which we design an adaptive scheme (hp-ADFEM) which hinges on a recent algorithm by P. Binev for adaptive hp-approximation. We prove convergence with contraction rate, and we investigate the optimality properties of the algorithm. This work is done in collaboration with R.H. Nochetto (College Park), R. Stevenson (Amsterdam), and M. Verani (Milan).