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In modern information technology, as integration density increases rapidly and the dimension of materials reduces to nanoscale, interfacial thermal transport (ITT) has attracted widespread attention of scientists. This review introduces the latest theoretical development in ITT through one-dimensional (1D) atomic junction model to address the thermal transport across an interface. With full consideration of the atomic structures in interfaces, people can apply the 1D atomic junction model to investigate many properties of ITT, such as interfacial (Kapitza) resistance, nonlinear interface, interfacial rectification, and phonon interference, and so on. For the ballistic ITT, both the scattering boundary method (SBM) and the non-equilibrium Green’s function (NEGF) method can be applied, which are exact since atomic details of actual interfaces are considered. For interfacial coupling case, explicit analytical expression of transmission coefficient can be obtained and it is found that the thermal conductance maximizes at certain interfacial coupling (harmonic mean of the spring constants of the two leads) and the transmission coefficient is not a monotonic decreasing function of phonon frequency. With nonlinear interaction—phonon–phonon interaction or electron–phonon interaction at interface, the NEGF method provides an efficient way to study the ITT. It is found that at weak linear interfacial coupling, the nonlinearity can improve the ITT, but it depresses the ITT in the case of strong-linear coupling. In addition, the nonlinear interfacial coupling can induce thermal rectification effect. For interfacial materials case which can be simulated by a two-junction atomic chain, phonons show interference effect, and an optimized thermal coupler can be obtained by tuning its spring constant and atomic mass.