<p>The increasing occupation of space orbits by fleets of satellites has led to increasing generation of data in space. At the same time, the expansion of technologies such as artificial intelligence (AI) has led to an increasing number of energy-intensive data centres, which have large carbon footprints, back on Earth. The former calls for space-based computing solutions, whereas the latter calls for carbon-neutral computing solutions. Here we explore the potential of developing carbon-neutral data centres in space. Such an approach would be enabled by the sustainability features of space: abundant solar energy that can be captured with high-efficiency solar cells and a giant cold heat sink (deep space itself) that can spontaneously take in large amounts of waste heat released from computing. We outline a framework for orbital edge data centres, which would be equipped with data sensors and AI accelerators, for carbon-neutral data processing at source in space. We then outline a framework for orbital cloud data centres in the form of a constellation of computational satellites equipped with servers and broadband connectivity, for both in-space and ground-outsourced computing applications. We also provide a method to evaluate the lifecycle carbon usage effectiveness of these cloud data centres.</p>