Differential complexes encode important algebraic and differential structures of physics models. Different problems involve different differential structures and complexes. For grad-div-curl related problems, such as those from electromagnetism and fluid dynamics,the de Rham complex plays a fundamental role. For other problems, such as those fromcontinuum mechanics, differential geometry, and general relativity, other complexes arerequired, such as the so-called elasticity (Kröner, Calabi) complex. These complexes andtheir properties can be systematically derived from the de Rham complex via a Bernstein–Gelfand–Gelfand (BGG) construction. There appears to be a neat correspondence between a large class of continuum mechanics models and the BGG machinery. Hence, differential complexes also provide a new angle for developing mechanics models and shed light on their structure-aware formulation. In this talk, we discuss the BGG machinery and their correspondence to elasticity, microstructures (micropolar models), continuum defects, dimension reduction, and multi-dimensional models. This paves a way for structure-preserving discretization.