As space agencies are planning manned missions to reach Mars in the next decades, researchers need to pave the way for supporting astronauts during their sojourn. The achievement of such an ambitious objective should not ignore the big issue of the substantial lack of wireless connectivity on the Mars surface. The landing of astronauts will impose the presence in situ of some basic supporting infrastructures. Unfortunately, the wireless infrastructures actually present on the Red planet are targeted at servicing sensor networks and not at providing useful connectivity to a human crew. In this chapter, we shall review the solutions proposed in the literature to bring broadband mobile connectivity on the Mars surface, considering also the possibility of exploiting the most advanced networking techniques that will be adopted on the Earth in the framework of 5G and beyond 5G standardization.