What is a probabilistically defined quantum field theory?

A mathematically precise definition of quantum field theories is a major and mostly unsolved goal of mathematical physics. One possible route to reaching this goal is through the program of constructive field theory, which is probabilistic at its core. This is one reason why there has been a flurry of activities in the probability community in the last ten years on Yang-Mills and other quantum field theories. I will try to give an overview of the constructive QFT program, its achievements, and the main open questions. I will also mention problems that lie beyond the scope of the probabilistic program, such as those coming from de Sitter quantum gravity, and briefly talk about my recent paper on this.