1. advance your position or become more successful
He/she is really moving up in the world
This use of this expression is not limited to any one facet of somebody's life, it can be used to refer to success in every aspect. For example, you can 'move up in the world' by advancing your social, professional, political, or financial position. For that reason, it is common in both a social and professional context. Two similar expressions are 'get ahead' or 'rise up the ranks.'