This paper presents a review of the state of the art for digital twins in the application domain of engineering dynamics. The focus on applications in dynamics is because: (i) they offer some of the most challenging aspects of creating an effective digital twin, and (ii) they are relevant to important industrial applications such as energy generation and transport systems. The history of the digital twin is discussed first, along with a review of the associated literature; the process of synthesizing a digital twin is then considered, including definition of the aims and objectives of the digital twin. An example of the asset management phase for a wind turbine is included in order to demonstrate how the synthesis process might be applied in practice. In order to illustrate modeling issues arising in the construction of a digital twin, a detailed case study is presented, based on a physical twin, which is a small-scale three-story structure. This case study shows the progression toward a digital twin highlighting key processes including system identification, data-augmented modeling, and verification and validation. Finally, a discussion of some open research problems and technological challenges is given, including workflow, joints, uncertainty management, and the quantification of trust. In a companion paper, as part of this special issue, a mathematical framework for digital twin applications is developed, and together the authors believe this represents a firm framework for developing digital twin applications in the area of engineering dynamics.