Abstract: The lattice vibrations (phonon modes) of crystals underpin a large number of material properties. The harmonic phonon spectrum of a solid is the simplest description of its structural dynamics and can be straightforwardly derived from the Hellman–Feynman forces obtained in a ground-state electronic structure calculation. The presence of imaginary harmonic modes in the spectrum indicates that a structure is not a local minimum on the structural potential-energy surface and is instead a saddle point or a hilltop, for example. This can in turn yield important insight into the fundamental nature and physical properties of a material. In this review article, we discuss the physical significance of imaginary harmonic modes and distinguish between cases where imaginary modes are indicative of such phenomena, and those where they reflect technical problems in the calculations. We outline basic approaches for exploring and renormalising imaginary modes, and demonstrate their utility through a set of three case studies in the materials sciences.