Many of the building blocks of life such as amino acids and nucleotides are chiral, i.e., different from their mirror image. Contemporary life selects and synthesizes only one of two possible handednesses. In an abiotic environment, however, there are usually equally many left- and right-handed molecules. If homochirality was a prerequisite of life, there must have been physical or chemical circumstances that led to the selection of a certain preference. Conversely, if it was a consequence of life, we must identify possible pathways for accomplishing a transition from a racemic to a homochiral chemistry. After a discussion of the observational evidence, we review ideas where homochirality of any handedness could emerge as a consequence of the first polymerization events of nucleotides in an emerging RNA world. These mechanisms are not limited to nucleotides, but can also occur for peptides, as a precursor to the RNA world. The question of homochirality is, in this sense, intimately tied to the origin of life. Future Mars missions may be able to detect biomolecules of extant or extinct life. We therefore also discuss possible experimental setups for determining the chirality of primitive life forms in situ on Mars.