Eutectic and peritectic solidification patterns

•Microstructure formation in eutectic and peritectic alloys is reviewed.•The pattern formation is interpreted with the tools of nonlinear physics.•Recent advances in in situ experiments are highlighted.•Recent advances in phase-field simulations are highlighted. Recent advances in the understanding of eutectic and peritectic two-phase pattern formation under purely diffusive transport are reviewed. The parallel progress of two key techniques, namely, in situ experimentation with model, low-melting transparent and metallic alloys in thin and bulk samples, and numerical phase-field simulations, is highlighted. Experiments and simulations are interpreted in the light of the theory of non-equilibrium pattern formation phenomena. Focus is put on microstructure selection and morphological transitions, multiscale patterns in ternary alloys, and the influence of crystallographic effects on pattern formation. Open problems, for example on crystallographic effects, irregular eutectics, and peritectic solidification, are outlined.