This paper investigates dynamical systems arising from theaction bytranslations on the orbit closures of self-similar andself-affine tilingsof ${\Bbb R}^d$. The main focus is on spectral properties of suchsystemswhich are shown to be uniquely ergodic. We establishcriteria forweak mixing and pure discrete spectrum for wide classes ofsuch systems.They are applied to a number of examples which includetilings withpolygonal and fractal tile boundaries; systems with purediscrete,continuous and mixed spectrum.