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Coulson's bond order is a chemically intuitive quantity that measures the difference in the occupation of bonding and anti-bonding orbitals. Both empirical and rigorously derived bond order expressions have evolved in the course of time and proven very useful for atomistic modeling of materials. The latest generation of empirical formulations has recently been augmented by screening-function approaches. Using friction and wear of diamond and diamond-like carbon as examples, we demonstrate that such a screened bond order scheme allows for a faithful description of dynamical bond-breaking processes in materials far from equilibrium. The rigorous bond order expansions are obtained by systematic coarse-graining of the tight binding approximation and form a bridge between the electronic structure and the atomistic modeling hierarchies. They have enabled bottom-up derivations of bond order potentials for covalently bonded semiconductors, transition metals, and multicomponent intermetallics. The recently developed magnetic bond order potential gives a correct description of both directional covalent bonds and magnetic interactions in iron and is able to correctly predict the stability of bulk Fe polymorphs as well as the intricate properties of dislocation cores. The bond order schemes hence represent a family of reliable and powerful models that can be applied in large-scale simulations of complex processes involving fracture, wear, and plasticity.