Eutectic phase transitions play an important role in many engineering materials, from cast iron to electronic solder. Regular binary eutectics are relatively straightforward and generally well understood, but the additional degree of freedom in three-component alloys introduces a far greater level of complexity, as three solid phases can form simultaneously from the melt. These ternary eutectic structures show promise for creating materials with unique optical and electronic properties, and are also useful for understanding the factors that control multi-phase, multi-component solidification more generally. The fundamentals of higher-order eutectics will be explained, before describing recent experimental and simulation work primarily on the model system of Al-Ag-Cu. By directional solidification at relatively low velocities and thermal gradients, ternary microstructures with varying degrees of alignment were produced and studied. These microstructures will be discussed both qualitatively and quantitatively, along with the role of diffusion, interfacial energy and crystal structure in their formation.