The flight of insects has fascinated scientists for centuries, and a large body of work has focused on understanding the intricacies of the aerodynamics of these organisms. Most of this work has focused on the flight of insects ranging in size from a fruit fly to a hawkmoth. The smallest flying insects, however, are ten times smaller than a fruit fly, and much less is known about their wing kinematics and aerodynamics. These insects fly at Reynolds numbers on the order of 10, and at this scale relative viscous forces are much greater than they are for larger insects. Consequently, these small insects must overcome significant aerodynamic challenges in order to propel themselves. The main goal of this dissertation is to better understand the challenges of flying at low Reynolds numbers and to investigate possible mechanisms to overcome those challenges. I used the immersed boundary method to solve for the fluid flow around insect wings in both two- and three-dimensions. In Chapter 3, I use a two-dimensional model to explore theoretically whether small insects could “swim” through the air using a drag-based stroke mechanism to generate vertical force. Chapter 4 investigates the aerodynamic role of bristled wings, especially engaged in wing-wing interactions. And finally, Chapter 5 introduces a three-dimensional revolving wing which is used to explore the forces and flow structures generated at low Reynolds numbers. Overall, the results suggest that there are drastic differences in the aerodynamics of flight at the scale of the smallest flying insects, which makes their flight more challenging, compared to larger insects. As the Reynolds number decreases, drag increases and the lift-to-drag ratio decreases significantly. Chapter 3 reveals that neither lift- nor drag-based vertical force generation mechanisms are very efficient at the range of the smallest flying insects. Chapter 4 demonstrates that bristled wings could reduce the force required to fling the wings apart during clap and fling while still maintaining lift during translation. And Chapter 5 shows that both the leading edge vortex and trailing edge vortex remain attached to the wing and spanwise flow decreases as the Reynolds number is lowered.