Peridynamics (PD) is arecently developed theory in solid mechanics that employs anonlocal model of force interaction, and replaces the partialdifferential equations, such as the stress-strain relationship ofthe classical continuum theory, by an integral operator that sumsup internal forces separated by finite distances. It allowsdiscontinuities of various types to be modeled without applicationof special techniques to the discontinuous field and shows apromising prospect for solving practical problems in whichdiscontinuities form and grow spontaneously. Peridynamics has beensuccessfully applied to damage and failure problems at both macro-and micro-scales with satisfactory solution precision andnumerical efficiency. Without pathological defects of traditionalmethods when facing discontinuous problems, and excluding thecomputational limitations in length and time scales, peridynamicsshows great potential analogous but advantageous to both classicalmeshfree and molecular dynamic methods. The present paper firstreviews its theoretical basis, numerical scheme and modelingmethod, and then elucidates its application to discontinuousproblems at different scale, including damage, fracture, impact,penetration and stability analysis for macro-scale homogeneous andheterogeneous materials and structures, kinetics analysis forphase transformations and atomistic analysis for nanoscalematerials. Finally, some unsolved problems and future researchtrends in PD are discussed.