We consider the scattering of time-harmonic acoustic waves by planar sound-soft (Dirichlet) and sound-hard (Neumann) screens. Such problems are classical when the screen is a Lipschitz open subset of the plane. But when this Lipschitz assumption fails (e.g. if the screen is fractal or has a fractal boundary) the classical formulations may be ill posed. Such problems arise naturally in models of fractal antennas in electrical engineering, and of light scattering by snowflakes and ice crystals in atmospheric physics. We present novel well-posed boundary value problem and integral equation formulations valid for arbitrary screens. The resulting solutions exhibit interesting behaviours - for instance, the scattering effect of a sound-soft screen depends critically on its ``capacity'', which is closely related to its fractal (Hausdorff) dimension. Our analysis relies on a careful study of fractional Sobolev spaces on non-Lipschitz domains, and the associated potential theory. Numerical computation and analysis of such problems poses many interesting challenges, and some preliminary numerical results will be reported.