  The first step in the construction of a regression model or a data-driven
analysis framework for matter at the atomic scale involves transforming the
Cartesian coordinates that describe the positions of the atoms in the form of a
representation that obeys the same symmetries as the properties of interest,
and in general reflects the physical nature of the problem. The link between
properties, structures, their physical chemistry and their mathematical
description is strongest when it comes to applications aimed at determining a
precise correspondence between atomic configurations and the quantities that
one might compute by a quantum mechanical electronic-structure calculation or
measure experimentally. The development of atomic-scale representations have
played, and continue to play, a central role in the success of machine-learning
methods that rely on this correspondence, such as interatomic potentials, as
well as generic property models, structural classifiers and low-dimensional
maps of structures and datasets. This review summarizes the current
understanding of the nature and characteristics of the most commonly used
structural and chemical descriptions of molecules and materials, highlighting
the deep underlying connections between different frameworks, and the ideas
that lead to computationally efficient and universally applicable models. It
gives 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.
