Fluid mixing is a challenging problem in laminar flow systems. Even in microfluidic systems, diffusion is often negligible compared to advection in the flow. The idea of chaotic advection can be applied in these systems to enhance mixing efficiency. Topological chaos can also lead to efficient and rapid mixing. In this dissertation, an approach to enhance fluid mixing in laminar flows without internal rods is demonstrated by using the idea of topological chaos. Periodic motion of three stirrers in a two-dimensional flow can lead to chaotic transport of the surrounding fluid. For certain stirrer motions, the generation of chaos is guaranteed solely by the topology of that motion and continuity of the fluid. This approach is in contrast to standard techniques. Appropriate stirrer motions are determined using the Thurston-Nielsen classification theorem, which also predicts a lower bound on the complexity of the dynamics in the flow. Work in this area has focused largely on using physical rods as stirrers, but the theorem also applies when the ``stirrers'' are passive fluid particles. In this thesis, topological chaos is theoretically and numerically investigated in lid-driven cavities and rectangular channels without internal rods. When a two-dimensional incompressible Newtonian flow is in the Stokes flow regime, the stream function satisfies the two-dimensional biharmonic equation. When the flow occurs in a lid-driven cavity with solid side walls, this equation can be solved using a method that is similar to the traditional Fourier expansion but uses an asymptotic approximation for the sum of high order terms. When the flow occurs between two infinite plates with spatially periodic boundary conditions, an exact solution in a rectangle with finite width, which represents the flow in this infinitely-wide cavity, can be obtained by using the principle of superposition. A fully developed, three-dimensional flow in a rectangular channel can be decomposed into an unperturbed Poiseuille flow in the axial direction and a lid-driven cavity secondary flow in the cross section. This model can be applied to numerically simulate either pressure-driven flow in a rectangular channel with surface grooves or electro-osmotic flow in a rectangular channel with variations in surface potential. In this dissertation, the occurrence of topological chaos in unsteady two-dimensional flows as well as steady three-dimensional flows without internal rods is demonstrated. For appropriate choices of boundary velocity on the top and/or bottom walls, there exist three periodic points in the flows that produce a chaos-generating motion. In steady flow through a three-dimensional rectangular channel, the axial direction plays the role of time and the periodic points lie on streamtubes that "braid" the surrounding fluid as it moves through the duct. When appropriate motion is applied on the boundary of the wide cavity or channel, topological chaos can also be generated in the flow. The stretching rate of non-trivial material lines in all these flows agrees with the prediction of the lower bound of topological entropy provided by the Thurston-Nielsen theorem. Ghost rod structures are found and analyzed in the lid-driven cavity and rectangular channel flows with solid side walls. The results suggest that the no-slip boundary condition on the stationary internal surfaces is one of the reasons for poor mixing in steady laminar three-dimensional flows considered previously with solid braided internal rods.Ph. D.