The world is entering a new era of exploring and exploiting outer space. The revolution in small, low-cost satellites, the recent initiatives from some countries to establish a legal framework, the increasing demand for technology metals and advances in space additive manufacturing have renewed the interest in space mining. Biomining, the use of microorganisms to extract and recover valuable metals from minerals and wastes, could be used as alternative ISRU technology for harnessing space resources. This paper reviews in situ resources available on the Moon, Mars, and Near-Earth Asteroids (NEAs) for implementing biomining processes in space, the effects of the space environment on biomining microbes, and space-based bioreactor designs that will enable leaching of metals from regoliths. A comparison between terrestrial and space biomining will also be presented, focusing on the differences in the composition of minerals on Earth and space, the types of microorganisms used for leaching, and the parameters that need to be optimised in the space biomining processes. Next steps to mature biomining approaches by combining knowledge from synthetic biology, systems biology, geomicrobiology and process engineering for space applications will also be explored. Through an integrative effort of these fields, biomining processes commonly employed on Earth can be harnessed for sustainable space exploration.