<p>Digital twins have been the subject of significant and growing interest across scientific, engineering, and medical domains. And yet the promise of digital twins relies critically on the availability of efficient and accurate computational models, as all of the computations required for and from digital twins are outer-loop/many-query analyses: digital twin synchronization based on data uses iterative algorithms or ensemble methods for inference and estimation; real-time decision-making with digital twins is based on many-query computations for optimization and control, sometimes under uncertainty, which exacerbates the many-query nature of the computation. These computations must happen in or near real time to be useful. However, many computational models for engineering systems arise from the discretization of governing partial differential equations on high-dimensional computational meshes, making the resulting models expensive to both store and evaluate. To realize the promise of digital twins, inexpensive and fit-for-purpose <i>reduced-order models</i> offer much promise, due to their success across many areas of engineering over the past half-century. The goal of this survey is to provide the engineering community with an overview of the state of the art in reduced-order modeling with an eye towards promise for digital twin applications. We will survey state approximation methods, model reduction based on dynamical systems and control theory, as well as goal-oriented model reduction methods accounting for a variety of additional demands, such as structure preservation, suitability for inverse problems, stability, and others.</p>