The paper reviews methods for representing atomic neighborhoods in computational chemistry and condensed matter physics, focusing on their faithfulness and suitability for fitting potential energy surfaces. The authors emphasize the importance of invariance to symmetries such as rotation, reflection, translation, and permutation of atoms. They demonstrate that commonly used descriptors, despite appearing different, are specific cases of a general approach using a set of basis functions with increasing angular wave numbers to expand the atomic neighborhood density function. The expansion requires higher wave numbers as the number of neighbors increases to achieve faithful representation, and variants of descriptors converge at varying rates. A new approach, Smooth Overlap of Atomic Positions (SOAP), is proposed to directly define similarity between atomic neighborhoods, showing it is closely related to invariant descriptors. The performance of various representations is tested by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.The paper reviews methods for representing atomic neighborhoods in computational chemistry and condensed matter physics, focusing on their faithfulness and suitability for fitting potential energy surfaces. The authors emphasize the importance of invariance to symmetries such as rotation, reflection, translation, and permutation of atoms. They demonstrate that commonly used descriptors, despite appearing different, are specific cases of a general approach using a set of basis functions with increasing angular wave numbers to expand the atomic neighborhood density function. The expansion requires higher wave numbers as the number of neighbors increases to achieve faithful representation, and variants of descriptors converge at varying rates. A new approach, Smooth Overlap of Atomic Positions (SOAP), is proposed to directly define similarity between atomic neighborhoods, showing it is closely related to invariant descriptors. The performance of various representations is tested by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.