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We study lattices in non‐positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through super‐rigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterized amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries.

Isometry groups of non‐positively curved spaces: discrete subgroups