The construction of structures with multiple interfaces and dielectric/magnetic heterostructures enables the design of materials with unique physical and chemical properties, which has aroused intensive interest in scientific and technological fields. Especially, for electromagnetic (EM) wave absorption, enhanced interface polarization and improved impedence match with high Snoek's limitation could be achieved by multiple interfaces and dielectric/magnetic heterostructures, respectively, which are benificial to high-efficiency electromagnetic wave absorption (EWA). However, by far, the principles in the design or construction of structures with multiple interfaces and dielectric/magnetic heterostructures, and the relationships between those structures or heterostructures and their EWA performance have not been fully summarized and reviewed. This article aims to provide a timely review on the research progresses of high-efficency EM wave absorbers with multiple interfaces and dielectric/magnetic heterostructures, focusing on various promising EWA materials. Particularly, EM attenuation mechanisms in those structures with multiple interfaces and dielectric/magnetic heterostructures are discussed and generalized. Furthermore, the changllenges and future developments of EM wave absorbers based on those structures are proposed.