There are at least two types of relevant critical points inside turbulent fluid flows: Instantaneous stagnation points where the instantaneous fluctuating velocity is zero and points where the fluid acceleration is instantaneously zero. Statistically, the former move slowly compared to the turbulent velocity fluctuations whereas the latter are swept with them. They also have a multiscale spatial structure which, combined with their temporal structure, can be shown to control turbulent diffusion of passive fluid element pairs and turbulent clustering of small and heavy particles/droplets. These conclusions are illustrated and validated using direct numerical simulations of 2D and 3D homogeneous isotropic turbulence.