Automorphism groups of structures endowed with the topology generated by stabilisers of small subsets are topological groups and indeed when countable they are Polish. In this talk I will discuss different group topologies on automorphism groups of homogeneous structures and especially focus on minimal group topologies. A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. I will provide some background, and discuss several classification of group topologies coarser than so called point-wise convergence topology in the case of automorphism groups of the countable random graph and Urysohn space. This is a joint work with Javier de la Nuez González.