Small satellite communications recently entered a period of massive interest driven by the uprising space applications. CubeSats are particularly attractive due to their low development costs which makes them very promising in playing a central role in the global wireless communication sector with numerous applications. Moreover, constellations of CubeSats in low-earth orbits can meet the increasing demands of global-coverage flexible low-cost high-speed connectivity. However, this requires innovative solutions to overcome the significant challenges that face high-data-rate low-power space communications. This paper provides a comprehensive and critical review of the design and architecture of recent CubeSat communication systems with a particular focus on their baseband architectures. The literature is surveyed in detail to identify all baseband design, testing, and demonstration stages as well as accurately describe the systems architecture and communication protocols. The reliability, performance, data rate, and power consumption of the reviewed systems are critically evaluated to understand the limitations of current CubeSat systems and identify directions of future developments. It is concluded that CubeSat communication systems still face many challenges, namely the development of energy-efficient high-speed modems that satisfy CubeSats requirements. Nevertheless, there are several promising directions for improvements such as the use of improved coding algorithms, use of Field Programmable Gate Arrays, multiple access techniques, beamforming, advanced antennas, and transition to higher frequency bands. By providing a concrete summary of current CubeSat communication systems and by critically evaluating their features, limitations, and offering insights about potential improvements, the review should aid CubeSat developers to develop more efficient and high data rate systems.