Abstract: In the field of numerical simulation of compressible flows, significant progress has been made in high-resolution shock-capturing schemes. But there are still many numerical challenges in the numerical simulation of compressible multi-material flows involving material interfaces, which are mainly manifested by excessively numerical dissipation and non-physical oscillations across interfaces. The discontinuity of fluid properties at the interface is the main reason for the difficulty of physical modelling and numerical simulating of compressible multi-material flows. To establish an efficient and high-order numerical algorithm for such flows, five aspects must be considered simultaneously, i.e., the numerical framework, the compatible discretization of non-conservative equations, the high-order bounded scheme, the interface-compression, and the multi-region calculation. In this paper we review our works in recent years from these aspects. Through the above-mentioned work, a class of high-order Eulerian numerical simulation methods with low dissipation and essentially non-oscillatory property has been established and successfully applied to interfacial instability and turbulent mixing of compressible multi-material fluids. These numerical methods have been integrated into numerical simulation softwares for engineering problems in weapon physics, and have been providing important technical support for related engineering tasks.