  The first step in the construction of a regression model or a data-driven
analysis, aiming to predict or elucidate the relationship between the atomic
scale structure of matter and its properties, involves transforming the
Cartesian coordinates of the atoms into a suitable representation. The
development of atomic-scale representations has played, and continues to play,
a central role in the success of machine-learning methods for chemistry and
materials science. This review summarizes the current understanding of the
nature and characteristics of the most commonly used structural and chemical
descriptions of atomistic structures, highlighting the deep underlying
connections between different frameworks, and the ideas that lead to
computationally efficient and universally applicable models. It emphasizes the
link between properties, structures, their physical chemistry and their
mathematical description, provides examples of recent applications to a diverse
set of chemical and materials science problems, and outlines the open questions
and the most promising research directions in the field.
