Nanowires with Rashba spin-orbit coupling represent a promising platform for the realization of one-dimensional topological superconductivity and Majorana bound states. In this work we investigate Majorana bound states in hybrid normal-superconductor and short superconductor-normal-superconductor junctions based on nanowires with Rashba spin-orbit coupling. In particular, we explore consequences of the topological phase transition as well as the non-locality and self conjugation properties of the Majorana states on the low-energy spectrum and the Josephson effect in the case of superconductor-normal-superconductor junctions. Our work shows the great potential of hybrid junctions as a platform for the study of topological superconductivity and Majorana bound states.