Instability-driven turbulence is ubiquitous in astrophysical and laboratory plasmas, where it is an important component to how these systems transport energy, momentum, particles, etc. This thesis is concerned with how two instabilities drive and interact with magnetohydrodynamic turbulence. The magnetorotational instability (MRI) is the best candidate for driving turbulence in well-ionized accretion disks. In this thesis, the turbulence driven by the MRI is studied with a particular focus on how it compares to strong, driven, incompressible MHD turbulence. High-resolution, high- Reynolds number setups are analyzed to determine the existence and char- acter of an inertial range of scales where a nonlinear cascade dominates the dynamics. In contrast to previous studies, systems with an imposed magnetic field— that activate the linear MRI—provide evidence for the existence of an inertial range when one considers the dynamics perpendicular to a strong, large-scale axial magnetic field that develops in the system. The outer scale of the turbulence is determined by balance between the linear shear, present at all scales, and the turbulent shear. In the case of a system without an imposed magnetic field—where the MRI dynamo is subcritical—evidence is found for self-sustained turbulence at magnetic Prandtl number Pm = 1. Previous work was not able produce such turbulence for systems at lower Reynolds numbers. A turbulent state is found to be easier to self-sustain in these systems in high-aspect-ratio domains, with angular momentum transport also highly dependent on aspect ratio. Vertically-extended domains exhibit higher transport. Azimuthally- extended domains show increased transport until a certain aspect ratio, beyond which the transport decreases. A phenomenological explanation is proposed by which the separation of toroidal magnetic flux vertically allows for increased transport and drive of the subcritical MRI dynamo; as the toroidal dimension is extended, the tearing mode acts to break up such large scale flux, reducing transport. Additionally, two-dimensional MHD turbulence is studied in order to test the predictions of new theories that hypothesize competition between the tearing mode and turbulent fluctuations in flows with high Lundquist number. A method is introduced to simulate evolution of single, critically- balanced eddy, thereby allowing the direct investigation of this competition. Results demonstrate the disruption of the correlated eddy structure on the shorter of the Alfvén or tearing time. Steepening of the energy spectrum is observed, while decreased alignment is not. Generally, these results support the proposed theory, with further investigation warranted.