When metamaterial structures meet functional materials, what will happen? The recent rise of the combination of metamaterial structures and functional materials opens new opportunities for dynamic manipulation of terahertz wave. The optical responses of functional materials are greatly improved based on the highly-localized structures in metamaterials, and the properties of metamaterials can in turn be manipulated in a wide dynamic range based on the external stimulation. In the topical review, we summarize the recent progress of the functional materials-based metamaterial structures for flexible control of the terahertz absorption and polarization conversion. The reviewed devices include but are not limited to terahertz metamaterial absorbers with different characteristics, polarization converters, wave plates, and so on. We review the dynamical tunable metamaterial structures based on the combination with functional materials such as graphene, vanadium dioxide (VO2) and Dirac semimetal (DSM) under various external stimulation. The faced challenges and future prospects of the related researches will also be discussed in the end.