Fracture and strength in atomic components are governed by mechanical instabilities at atomic scales associated with irreversible deformations through bond breaking/switching, such as cleavage, dislocation nucleation, and phase transformations of crystal lattices. It is therefore of central importance to determine the critical conditions where atomic structures becomes mechanically unstable. Here we review the state-of-the-art theory for “fracturemechanics of atomic structures” that provides a rigorous description of mechanical instabilities in arbitrary atomic structures under any external loading/constraint. The theory gives the critical instability condition by positivity of the minimum eigenvalue of the Hessian matrix of the total energy with respect to degrees of freedom of the system (i.e., instability criterion), and it successfully provides atomistic insights into fracture in various atomic/nanoscale structures. The review also covers the recent development of theory extended to advanced systems including large-scale, finite temperature, and “multi-physics” instabilities in (ferro-)electric and magnetic materials as functional fracture.