This paper reviews recent methods for representing atomic environments and analyzes their effectiveness in fitting potential energy surfaces. The key properties of such representations are differentiability with respect to atomic movements and invariance to physical symmetries like rotation, reflection, and translation. The authors demonstrate that some widely used descriptors are specific cases of a general approach using basis functions with increasing angular wave numbers to expand the atomic neighborhood density function. They show that higher wave numbers are needed for accurate representation as the number of neighbors increases, and that different descriptors converge at varying rates. A new approach, Smooth Overlap of Atomic Positions (SOAP), is proposed, which directly defines the similarity between atomic environments and 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 bulk crystals. The paper also discusses other descriptors, including bond-order parameters, the power spectrum, the bispectrum, radial basis functions, and angular Fourier series. It concludes that the choice of descriptors and similarity measures is critical for obtaining accurate potential energy surfaces.This paper reviews recent methods for representing atomic environments and analyzes their effectiveness in fitting potential energy surfaces. The key properties of such representations are differentiability with respect to atomic movements and invariance to physical symmetries like rotation, reflection, and translation. The authors demonstrate that some widely used descriptors are specific cases of a general approach using basis functions with increasing angular wave numbers to expand the atomic neighborhood density function. They show that higher wave numbers are needed for accurate representation as the number of neighbors increases, and that different descriptors converge at varying rates. A new approach, Smooth Overlap of Atomic Positions (SOAP), is proposed, which directly defines the similarity between atomic environments and 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 bulk crystals. The paper also discusses other descriptors, including bond-order parameters, the power spectrum, the bispectrum, radial basis functions, and angular Fourier series. It concludes that the choice of descriptors and similarity measures is critical for obtaining accurate potential energy surfaces.