Emergence of two-dimensional (2D) materials with atomic-layer structures such as graphene and MoS2, which have excellent physical properties, provide the opportunity of substituting silicon-based micro/nano-electronics. An important issue before large-scale applications is the heat dissipation performance of these materials, especially when they are supported on a substrate, as in most scenarios. Thermal transport across the atomic-layer interface is essential to the heat dissipation of 2D materials due to the extremely large contact area with the substrate, when compared with their atomic-scale cross-sections. Therefore, the understanding of the interfacial thermal transport is important, but the characterization is very challenging due to the limitations for temperature/thermal probing of these atomic-layer structures. In this review, widely used characterization techniques for experimental characterization as well as their results are presented. Emphasis is placed on the Raman-based technology for nm and sub-nm temperature differential characterization. Then we present physical understanding through theoretical analysis and molecular dynamics. A few representative works about the molecular dynamics studies, including our studies on the size effect and rectification phenomenon of the graphene-Si interfaces are presented. Challenges as well as opportunities in the thermal transport study of atomic-layer structures are discussed. Though many works have been reported, there is still much room in both the development of experimental techniques as well as atomic-scale simulations for a clearer understanding of the physical fundamentals of thermal transport across the atomic-layer interfaces, considering the remarkable complexity of physical/chemical conditions at the interface.