Protoplanetary disks play a key role in star and planet formation processes. Turbulence in these disks, which arises from the magnetorotational instability (MRI), not only causes accretion of mass onto the central star, but also sets the conditions for processes such as dust settling, planetesimal formation, and planet migration. However, the exact nature of this turbulence is still not very well constrained in these systems. In this talk, I describe a new, emerging paradigm for the nature of disk turbulence and accretion in the presence of a large scale magnetic field. I first present a series of high resolution numerical simulations of radially localized regions of accretion disks in order to understand the influence of low-ionization, non-ideal physics on the amplitude and nature of turbulence driven by the magnetorotational instability (MRI). I focus in particular on the role of the Hall effect and show that it introduces a bimodality in disk properties, depending on the orientation of the large scale magnetic field with respect to the angular momentum vector of the orbiting gas. I will then describe some recent work that I have carried out utilizing both state-of-the-art numerical simulations and powerful new ALMA observations, to directly link numerical predictions for the turbulent velocity structure of protoplanetary disks to observations. Preliminary results show that turbulence is surprisingly weak, and I conclude with some interpretations of these results.