Direct connectivity in L/S frequency bands between satellites and common mobile terrestrial user equipment (UE), such as smartphones, is an essential feature for future 6G non-terrestrial networks. The technical trend in closing the link between the communication endpoints is to develop large phased antenna arrays to be launched in LEO orbit. Satellite swarms represent an innovative and promising approach. Swarms are composed of several small and lightweight satellites organized in a free-flying formation (i.e., wireless connected) or a tethered formation (i.e., wired connected) creating a distributed phased antenna array. It has the potential to provide an enhanced gain, narrower beam width and lower launch/build costs compared to conventional single satellite systems with large phased antenna arrays. The first objective of this work is the design of swarm-based antenna arrays, in which the impact of key parameters such as the number of satellites in the swarm, their reciprocal distance and the array geometry, is thoroughly analyzed. It is shown that the undesired phenomenon of grating lobes can be mitigated via optimized array geometries and a new geometry named the enhanced logarithmic spiral array (ELSA) is presented. The second objective of this work is the identification of the most important research directions and system design aspects for the swarm system. In particular, it is shown that tethered swarms with ELSA geometries, innovative deployable structures and very small satellites can foster the deployment of swarms in future satellite systems.