The field of satellite communications is enjoying a renewed interest in the global telecom market, and very high throughput satellites (V/HTS), with their multiple spot-beams, are key for delivering the future rate demands. In this article, the state-of-the-art and open research challenges of signal processing techniques for V/HTS systems are presented for the first time, with focus on novel approaches for efficient interference mitigation. The main signal processing topics for the ground, satellite, and user segment are addressed. Also, the critical components for the integration of satellite and terrestrial networks are studied, such as cognitive satellite systems and satellite-terrestrial backhaul for caching. All the reviewed techniques are essential in empowering satellite systems to support the increasing demands of the upcoming generation of communication networks.