In this talk I will start by giving an introduction to the properties of quasicrystals and discuss how they differ from, say, disordered or fractal structures. Experimental results for electronic properties, described briefly, motivated studies of tight-binding models, which I will introduce. Results obtained for electron wavefunctions, quantum diffusion properties, the effect of disorder will be discussed along with proposals for experiments to observe some of the predicted singular properties of quasicrystals.