Schroedinger difference equation with deterministic ergodic potentials

We review the recent developments in the theory of the one-dimensional tight-binding Schr\"odinger equation for a class of deterministic ergodic potentials. In the typical examples the potentials are generated by substitutional sequences, like the Fibonacci or the Thue-Morse sequence. We concentrate on rigorous results which will be explained rather than proved. The necessary mathematical background is provided in the text.