This paper investigates dynamical systems arising from the action by translations on the orbit closures of self-similar and self-affine tilings of ${\Bbb R}^d$. The main focus is on spectral properties of such systems which are shown to be uniquely ergodic. We establish criteria for weak mixing and pure discrete spectrum for wide classes of such systems. They are applied to a number of examples which include tilings with polygonal and fractal tile boundaries; systems with pure discrete, continuous and mixed spectrum.