Abstract: Recently, due to the infinite design space, outstanding capability in changing shape, dimension, and topology, as well as the folding-induced extraordinary mechanical properties, origami structures and origami metamaterials have rapidly become the research frontiers and hot spots in the fields of mathematics, physics, and engineering. Origami structures and origami metamaterials have extensive application prospects in various fields, including aerospace, medical, and robotic engineering. Typical examples are the large-scale deployable aerospace structures, reconfigurable self-folding robots, and micro-scale foldable devices. As the scope of engineering applications continues to expand, the dynamics of origami structures and origami metamaterials become increasingly prominent, which not only involves dynamic modeling and parameter identification but also relates to the dynamic mechanism analyses and experimental tests. The origami dynamics research is facing many new challenges and opportunities brought by the complex spatial geometric relations, the rich deformation modes, and the folding-induced global strong nonlinear constitutive profiles. In this review, the research background and significance of origami structures and origami metamaterials are firstly surveyed, followed by a brief introduction to the fundamental definitions, assumptions, and categorization of origami. The geometric design, kinematic and static properties of the origami structures and origami metamaterials are also summarized in brief. Afterward, the recent research progress on the dynamics of origami structures and origami metamaterials are systematically reviewed, from the following aspects: (1) dynamic modeling and parameter identification methods; (2) theoretical, finite element, and experimental approaches for dynamic analysis; (3) folding-induced dynamic behaviors, including bi-stable and multi-stable dynamic behaviors, transient dynamic behaviors, and wave propagation dynamic behaviors, etc.; (4) typical dynamic applications. Finally, several open problems are addressed for future studies.