Stable subgroup of non-hyperbolic groups have been introduced by Durham and Taylor as a well defined notion of convexity in non-hyperbolic groups. For different examples of groups, stable subgroups are characterized through a well known action on a hyperbolic space. We will describe such characterizations of stable subgroups in RAAGs and mapping class groups and will see what are the analogs for graph products. This is based on joint work with S. H. Balasubramanya, M. Chesser, A. Kerr and J. Mangahas.