Kinematical Lie groups are generalisations of the relativity groups and include the group of galilean transformations, as well as the isometries of Minkowski and (anti)de Sitter spacetimes. We will review the recent classification of kinematical Lie algebras in arbitrary dimension and the classification of the corresponding homogeneous spacetimes. Most of these spacetimes do not have invariant metrics, but rather Newton-Cartan and/or carrollian structures, which we will review.