The macroscopic elasticity of a solid is a direct consequence of the interactions between particles at the atomic level. Accordingly, the elasticity of a crystal reflects the strength and anisotropy of its bonding system like no other property. Extreme values of longitudinal elastic stiffness often occur in directions of particularly strong or weak periodic bond chains. However, the definition of “strong periodic bond chain” is not fully compatible with the interpretation of elastic stiffnesses as spring constants. Qualitative differences in the interactions between the particles, in particular whether they are directional or non-directional, become clear in the so-called deviations from Cauchy relations. Even a quasi-additivity rule can be formulated for the elastic S-values, which allows the mean elastic stiffness of a new crystal species to be estimated. The possibilities and limitations of using these structure-property relationships are explained using selected examples.