Viscous fingering is a classical hydrodynamic instability that occurs when a fluid is injected into a porous medium or Hele-Shaw cell that already contains a more viscous fluid. The result is that the invading fluid will propagate through the defending fluid in narrow, finger-like channels rather than displacing it uniformly. As with most instabilities, viscous fingering can be desirable or undesirable---For example, it has a strong and adverse impact on enhanced oil recovery and many manufacturing processes, but it can also be exploited to promote mixing in microfluidic devices. In these and other applications, it would be extremely useful to be able to suppress, enhance, or otherwise control this phenomenon. The prospect of controlling hydrodynamic instabilities such as viscous fingering is a long-standing challenge across engineering. One promising approach for achieving this is by manipulating the flow with a moveable or deformable solid structure. Most previous efforts to control instabilities via such fluid-structure interactions (FSI) have been directed at inertial flows (e.g., aerodynamic drag and turbulence), but this idea also has clear relevance to viscous flows (e.g., in microfluidics, biomedical engineering, and subsurface flow), including viscous fingering. The goal of this project is to strengthen and broaden our understanding of the impact of FSI on viscous flows in general, and on viscous fingering in particular. We will first develop a novel apparatus for measuring and controlling the impact of FSI on viscous flow in a Hele-Shaw cell. Then, by combining high-resolution experiments in this system with mathematical modelling, we will develop a new strategy for controlling viscous fingering. This strategy for enhancing or suppressing viscous fingering can then be applied to a wide variety of practical challenges, from microfluidics to enhanced oil recovery.