Non-linear realisations of the groups SU(2) and SO(1,4) are analysed, described by the coset spaces SU(2)/U(1), SO(1,4)/SO(1,3). The analysis consists of determining the transformation properties of the Goldstone bosons, constructing the most general possible Lagrangian for the realisations and finding the metric of the coset space. The Lie algebras of special unitary groups are studied and their projection operators are determined, leading to a general method for constructing the Lagrangian for a non-linear realisation of a special unitary group. The Lie algebra of SU(4) is looked at in depth and its homomorphism with S0(6) allows a full specification of the most general Lagrangian for the coset space SO(6)/SO(4)⊗SO(2).