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Kaluza-Klein theory is developed starting from the simplest example in which a single extra spatial dimension is compactified to a circle, and a single Abelian gauge field emerges in four dimensions from the higher-dimensional metric. This is generalised to greater dimensionality whence non-Abelian gauge groups may be obtained, and possible mechanisms for achieving the compactification of the extra spatial dimensions are discussed. The spectrum of particles appearing in four dimensions isdoi:10.1088/0034-4885/50/9/001 fatcat:uoayerx2kzeudogg23ka4lucji