Abstract: Data-driven models that respect physical laws are robust to noise, require few training samples, and are highly generalisable. Although the dynamic mode decomposition (DMD) is a principal tool of data-driven fluid dynamics, it is rare for learned DMD models to obey physical laws such as symmetries, invariances, causalities, spatial locality and conservation laws. Thus, we present physics-informed dynamic mode decomposition (piDMD), a suite of tools that incorporate physical structures into linear system identification. Specifically, we develop efficient and accurate algorithms that produce DMD models that obey the matrix analogues of user-specified physical constraints. Through a range of examples from fluid dynamics, we demonstrate the improved diagnostic, predictive and interpretative abilities of piDMD. We consider examples from stability analysis, data-driven resolvent analysis, reduced-order modelling, control, and the low-data and high-noise regimes.