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.