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Wind farm control has been a topic of research for more than two decades. It has been identified as a core component of grand challenges in wind energy science to support accelerated wind energy deployment and transition to a clean and sustainable energy system for the 21st century. The prospect of collective control of wind turbines in an array to increase energy extraction, reduce loads, improve the balance of systems, reduce operation and maintenance costs, etc. has inspired many researchers over the years to propose innovative ideas and solutions. However, practical demonstration and commercialization of some of the more advanced concepts has been limited by a wide range of challenges, which include the complex physics of turbulent flows in wind farms and the atmosphere, uncertainties related to predicting load and failure statistics, and the highly multi-disciplinary nature of the overall design optimization problem, among others. In the current work, we aim at providing a comprehensive overview of the state of the art and outstanding challenges, thus identifying the key research areas that could further enable commercial uptake and success of wind farm control solutions. To this end, we have structured the discussion on challenges and opportunities into four main areas: (1) insight in control flow physics, (2) algorithms and AI, (3) validation and industry implementation, and (4) integrating control with system design (co-design).