Space–time adaptive and high-order methods for valuing American options using a partial differential equation (PDE) approach are developed in this paper. The linear complementarity problem that arises due to the free boundary is handled using a penalty method. Both finite difference and finite element methods are considered for the space discretization of the PDE, while classical finite differences, such as Crank–Nicolson, are used for the time discretization. The high-order discretization in space is based on an optimal finite element collocation method, the main computational requirements of which are the solution of one tridiagonal linear system at each timestep, while the resulting errors at the grid points and midpoints of the space partition are fourth order. To control the space error we use adaptive grid-point distribution based on an error equidistribution principle. A timestep size selector is used to further increase the efficiency of the methods. Numerical examples show that our methods converge fast and provide highly accurate options prices, Greeks and early exercise boundaries.