A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in the normed spaces (R d , · q ), for d ≥ 2 and 1 < q < ∞. Generalisations are obtained for the Laman and Henneberg combinatorial characterisations of generic infinitesimal rigidity for finite graphs in (R 2 , · 2). Also Tay's multi-graph characterisation of the rigidity of generic finite body-bar frameworks in (R d , · 2) is generalised to the non-Euclidean norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit G = lim − → G k of an inclusion tower of finite graphs G1 ⊆ G2 ⊆ . . . for which the inclusions satisfy a relative rigidity property. For d ≥ 3 a countable graph which is rigid for generic placements in R d may fail the stronger property of sequential rigidity, while for d = 2 the equivalence with sequential rigidity is obtained from the generalised Laman characterisations. Applications are given to the flexibility of non-Euclidean convex polyhedra and to the infinitesimal and continuous rigidity of compact infinitely-faceted simplicial polytopes.