The goal of seismic inversion is to obtain subsurface properties from surface measurements. Seismic images have proven valuable, even crucial, for a variety of applications, including subsurface energy exploration, earthquake early warning, carbon capture and sequestration, estimating pathways of sub-surface contaminant transport, etc. These subsurface properties~(such as wave speed, density, impedance, and reflectivity) influence the transmission of seismic waves through the subsurface media, and well-understood physics models (so-called “forward models”) can be used to predict what surface measurements would be made, for any given subsurface configuration. Seismic inversion is the inverse problem: given actual surface measurements, infer what subsurface configuration would give rise to those measurements. Like most inverse problems, seismic imaging is ill-posed. There is more below the surface than on the surface, and many different subsurface configurations can give rise to the same surface measurements. Because the forward model is itself computationally expensive – the inverse inference is even more so. But recent advances in algorithms and computing provide an opportunity for remarkable progress in seismic inversion, and efficient solutions to previously infeasible problems have been obtained using data-driven approaches (such as the deep learning methods that were developed primarily for problems in computer vision). The excellent performance of learning-based methods arises from its ability to exploit large amounts of high-quality training data, without the need for hand-designed features. Unlike computer vision, however, seismic inversion is not a data-rich domain. The technical challenges and high cost of acquiring field data prevent the accumulation of large quantities of data (and the commercial value of that data, once acquired) prevents much of the existing data from being widely available. To alleviate the data scarcity issue and improve model generalization, there has been growing interest in combining physics knowledge with machine learning for solving seismic inversion problems. This review will survey methods for incorporating physics knowledge with machine learning (primarily deep neural networks) to solve computational seismic inversion problems. We will provide a structured framework of the existing research in the seismic inversion community, and will identify technical challenges, insights, and trends.