With Neil Trudinger,we study the Plateau problem for affine maximal hypersurfaces,which is the affine invariant analogue of the classical Plateau problem for minimal surfaces.We formulate the affine Plateau problem as a geometric variational problem for the affine area functional,and prove the existence and interior regularity of maximizers.As a special case,we obtain corresponding existence and regularity results for the variational Dirichlet problem for the fourth order affine maximal surface equation.Our approach makes use of techniques and results from the theory of the Monge-Ampère equation.