Large scale geometry of finitely generated or locally compact groups has long been one of the cornerstones of geometric group theory and its connections with harmonic and functional analysis. However, many of the groups of interest in logic, topology and analysis fail to be locally compact, such as automorphism groups of countable structures, diffeomorphism and isometry groups. For these there has been no canonical way of defining their large scale structure, as it is possible, e.g., with the word metric on a finitely generated group. Moreover, recently many groups have turned out to have no non-trivial large scale structure at all, despite being non-compact. We present a theory of large scale structure of metrisable groups and among other things determine the necessary and sufficient conditions for this structure to be unique up to coarse or quasi-isometric equivalences. Applications to model theory will be presented.