Signal processing played an important role in improving the quality of communications over copper cables in earlier DSL technologies. Even more powerful signal processing techniques are required to enable a gigabit per second data rate in the upcoming G.fast standard. This new standard is different from its predecessors in many respects. In particular, G.fast will use a significantly higher bandwidth. At such a high bandwidth, crosstalk between different lines in a binder will reach unprecedented levels, which are beyond the capabilities of most efficient techniques for interference mitigation. In this article, we survey the state of the art and research challenges in the design of signal processing algorithms for the G.fast system, with a focus on novel research approaches and design considerations for efficient interference mitigation in G.fast systems. We also detail relevant VDSL techniques and points out their strengths and limitations for the G.fast system.