The paper by Félix Mouhat and François-Xavier Coudert addresses the necessary and sufficient conditions for elastic stability in various crystal systems, focusing on cubic, hexagonal, tetragonal, rhombohedral, orthorhombic, monoclinic, and triclinic crystals. The authors aim to provide a concise and pedagogical reference for stability criteria in non-cubic materials, addressing the confusion in the literature regarding the form of these conditions for lower symmetry crystal classes. The elastic behavior of a crystal is described by a stiffness matrix, which has 21 independent components for arbitrary crystals. The stability conditions are derived from the requirement that the stiffness matrix be definite positive, all eigenvalues be positive, and all leading principal minors be positive. These conditions are equivalent to the Born elastic stability criteria. For cubic crystals, the conditions are well-known and simple. For hexagonal and tetragonal (I) classes, the conditions involve additional constraints due to the reduced number of independent elastic constants. For rhombohedral (I) and (II) classes, the conditions are more complex, involving inequalities that ensure the positivity of specific minors. For orthorhombic crystals, the conditions involve cubic polynomials in the elastic constants, while for monoclinic and triclinic systems, they involve quartic and sextic polynomials, respectively. The authors note that simpler conditions proposed in the literature for orthorhombic crystals are incorrect and only necessary but not sufficient. The paper concludes by discussing the generalization of these stability conditions to systems under external loads, where the stiffness tensor changes, and the resulting symmetry of the tensor may be lower than that of the original stiffness matrix.The paper by Félix Mouhat and François-Xavier Coudert addresses the necessary and sufficient conditions for elastic stability in various crystal systems, focusing on cubic, hexagonal, tetragonal, rhombohedral, orthorhombic, monoclinic, and triclinic crystals. The authors aim to provide a concise and pedagogical reference for stability criteria in non-cubic materials, addressing the confusion in the literature regarding the form of these conditions for lower symmetry crystal classes. The elastic behavior of a crystal is described by a stiffness matrix, which has 21 independent components for arbitrary crystals. The stability conditions are derived from the requirement that the stiffness matrix be definite positive, all eigenvalues be positive, and all leading principal minors be positive. These conditions are equivalent to the Born elastic stability criteria. For cubic crystals, the conditions are well-known and simple. For hexagonal and tetragonal (I) classes, the conditions involve additional constraints due to the reduced number of independent elastic constants. For rhombohedral (I) and (II) classes, the conditions are more complex, involving inequalities that ensure the positivity of specific minors. For orthorhombic crystals, the conditions involve cubic polynomials in the elastic constants, while for monoclinic and triclinic systems, they involve quartic and sextic polynomials, respectively. The authors note that simpler conditions proposed in the literature for orthorhombic crystals are incorrect and only necessary but not sufficient. The paper concludes by discussing the generalization of these stability conditions to systems under external loads, where the stiffness tensor changes, and the resulting symmetry of the tensor may be lower than that of the original stiffness matrix.