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This is a review of hadron structure physics from lattice QCD. Throughout this report, we place emphasis on the contribution of lattice results to our understanding of a number of fundamental physics questions related to, e.g., the origin and distribution of the charge, magnetization, momentum and spin of hadrons. Following an introduction to some of the most important hadron structure observables, we summarize the methods and techniques employed for their calculation in lattice QCD. We briefly discuss the status of relevant chiral perturbation theory calculations needed for controlled extrapolations of the lattice results to the physical point. In the main part of this report, we give an overview of lattice calculations on hadron form factors, moments of (generalized) parton distributions, moments of hadron distribution amplitudes, and other important hadron structure observables. Whenever applicable, we compare with results from experiment and phenomenology, taking into account systematic uncertainties in the lattice computations. Finally, we discuss promising results based on new approaches, ideas and techniques, and close with remarks on future perspectives of the field.