Multi-distribution learning is a natural generalization of PAC learning to settings with multiple data distributions. There remains a significant gap between the known upper and lower bounds for PAC-learnable classes. In particular, though we understand the sample complexity of learning a VC dimension d class on $k$ distributions to be $O(ε^{-2} \ln(k)(d + k) + \min\{ε^{-1} dk, ε^{-4} \ln(k) d\})$, the best lower bound is $Ω(ε^{-2}(d + k \ln(k)))$. We discuss recent progress on this problem and some hurdles that are fundamental to the use of game dynamics in statistical learning.