Metal additive manufacturing (AM) presents advantages such as increased complexity for a lower part cost and part consolidation compared to traditional manufacturing. The multiscale, multiphase AM processes have been shown to produce parts with non-homogeneous microstructures, leading to variability in the mechanical properties based on complex process–structure–property (p-s-p) relationships. However, the wide range of processing parameters in additive machines presents a challenge in solely experimentally understanding these relationships and calls for the use of digital twins that allow to survey a larger set of parameters using physics-driven methods. Even though physics-driven methods advance the understanding of the p-s-p relationships, they still face challenges of high computing cost and the need for calibration of input parameters. Therefore, data-driven methods have emerged as a new paradigm in the exploration of the p-s-p relationships in metal AM. Data-driven methods are capable of predicting complex phenomena without the need for traditional calibration but also present drawbacks of lack of interpretability and complicated validation. This review article presents a collection of physics- and data-driven methods and examples of their application for understanding the linkages in the p-s-p relationships (in any of the links) in widely used metal AM techniques. The review also contains a discussion of the advantages and disadvantages of the use of each type of model, as well as a vision for the future role of both physics-driven and data-driven models in metal AM.