Topological homogeneity is not a well understood notion. As far as we know, there is only one known ZFC example of a homogeneous compact space which is not a product of dyadic spaces and first countable spaces. We discuss several classes of homogeneous spaces, among them the countable dense homogeneous spaces and the uniquely homogeneous spaces. We also state some intriguing open problems, some of which are (very) old and some of which came from recent investigations.