- Trace maps for products of transfer matrices prove to be an important tool in the investigation of electronic spectra and wave functions of one-dimensional quasiperiodic systems. These systems belong to a general class of substitution sequences. In this work we review the various stages of development in constructing trace maps for products of (2×2) matrices generated by arbitrary substitution sequences. The dimension of the underlying space of the trace map obtained by means of this construction is the minimal possible, namely 3r-3 for an alphabet of size r>=2. In conclusion, we describe some results from the spectral theory of discrete Schrödinger operators with substitution potentials.