Over the past few years, the number of space-related applications has increased as the acceptance of higher risk thresholds in the space industry has facilitated the widespread integration of commercial off-the-shelf (COTS) components, leading to a transformation in the design of new space missions. One of the most attractive areas of current space research is the design of miniaturized satellites, known as Cubesats, which are tempting because of their numerous applications and low design and deployment cost. Cubesats are now widely used in numerous commercial, scientific and research applications as they offer a number of interesting features in terms of development and versatility. Cubesats also provide a complementary connectivity solution to existing networks and on-board computation plays a crucial role in the design of Cubesat missions, as requirements are needed to be able to provide communications payloads that meet the objectives. This article analyzes the types of on-board computers (OBC) and key features offered by Cubesat missions in projects of all kinds aimed at providing communications via this type of satellites. Finally, it identifies future research lines for Cubesat communications in relation to OBCs.