Explore the intricacies of fluid dynamics in this comprehensive lecture on fluid equations, regularity, and Kolmogorov's turbulence theory. Delve into the regularity theory for the Navier-Stokes equation and examine the motivations behind Kolmogorov's phenomenological theory of turbulence. Gain insights into rigorous mathematical results that confirm various phenomenologies. Cover topics such as critical space, conditional regularity results, classical regularity results, determining modes, intermittency, irregular situations, and magnetohydrodynamics. Learn from Mimi Dai, a von Neumann Fellow from the University of Illinois, Chicago, as she presents this Members' Colloquium at the Institute for Advanced Study.