Networking and Artificial Intelligence (AI) are two of the most transformative information technologies over the last few decades. Building upon the synergies of these two powerful technologies, we envision designing next generation of edge networks to be highly efficient, reliable, robust and secure. To this end, in this paper, we delve into interesting and fundamental research challenges and opportunities that span two major broad and symbiotic areas: AI for Networks and Networks for AI. The former deals with the development of new AI tools and techniques that can enable the next generation AI-assisted networks; while the latter focuses on developing networking techniques and tools that will facilitate the vision of distributed intelligence, resulting in a virtuous research cycle where advances in one will help accelerate advances in the other. A wide range of applications will be further discussed to illustrate the importance of the foundational advances developed in these two areas.