  This paper is about the tiling dynamical systems approach to the study of
aperiodic order. We compare and contrast four related types of systems:
ordinary (one-dimensional) symbolic systems, one-dimensional tiling systems,
multidimensional $Z^d$-systems, and multidimensional tiling systems.
Aperiodically ordered structures are often hierarchical in nature, and there
are a number of different yet related ways to define them. We will focus on
what we are calling "supertile construction methods": symbolic substitution in
one and many dimensions, S-adic sequences, self-similar and pseudo-self-similar
tilings, and fusion rules. The techniques of dynamical analysis of these
systems are discussed and a number of results are surveyed. We conclude with a
discussion of the spectral theory of supertile systems from both the dynamical
and diffraction perspectives.
