In this review article, we present recent studies of peculiar Rashba spin-split energy band structures appearing at surfaces of metallic and semiconducting substances as well as surfaces of topological insulators utilizing state-of-the-art spin- and angle-resolved photoemission spectroscopy. It is shown in this article that spin- and angle-resolved spectroscopy plays very important roles in clarifying the spin- split surface and bulk band structures of Rashba-type spin-split systems and topological insulators. This article starts with instructive descriptions of the Rashba effect and topological insulators, which is followed by the history and recent development of spin- and angle-resolved photoemission spectroscopy.