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This thesis studies rigidity of graphs and hypergraphs realised in homogeneous and locally homogeneous spaces. We develop an algebraic model for describing the motions of such realisations using group-theoretic methods. The resulting structures, which we call graph-of-groups realisations, provide a unified model capable of capturing a wide range of rigidity problems. Within this model, we recover foundational results, including a necessary condition for rigidity. Using homological methods, we show that, generically, this condition is also sufficient for rigidity problems in homogeneous spaces G/H, where H is a one-dimensional and selfnormalising subgroup of a Lie group G. The resulting combinatorial conditions can be verified using the pebble game algorithm, which we analyse in novel settings. Beyond homogeneous spaces, we also study realisations of graphs in locally homogeneous spaces by interpreting these graphs as symmetric graphs in a homogeneous space. In particular, we study the minimal rigidity of symmetric graphs in the hyperbolic plane and obtain a combinatorial characterisation of minimal rigidity of graphs realised on compact orientable surfaces of genus g≥2.