Turbulence is ubiquitous in naturally-occurring and man-made flows. Despite its importance in scientific and engineering applications, the transition from smooth laminar flow to disorganized turbulent flow is poorly understood. In some cases, the transition can be understood in the context of linear stability theory, which predicts when the underlying laminar solution will become unstable as a parameter is varied. For a large class of flows, however, this approach fails spectacularly, with theory predicting that the laminar flow is stable but experiments and simulations showing the emergence of spatiotemporal complexity. In this dissertation, the direct or subcritical transition to turbulence in Taylor-Couette flow (i.e., the flow between independently rotating co-axial cylinders) is studied experimentally. Chapter 1 discusses different scenarios for the transition to turbulence and recent advances in understanding the subcritical transition within the framework of dynamical systems theory. Chapter 2 presents a comprehensive review of earlier investigations of linearly stable Taylor-Couette flow. Chapter 3 presents the first systematic study of long-lived super-transients in Taylor-Couette flow with the aim of determining the correct dynamical model for turbulent dynamics in the transitional regime. Chapter 4 presents the results of experiments regarding the stability of Taylor-Couette flow to finite-amplitude perturbations in the form of injection/suction of fluid from the test section. Chapter 5 presents numerical investigations of axisymmetric laminar states with realistic boundary conditions. Chapter 6 discusses in detail the implementation of time-resolved tomographic particle image velocimetry (PIV) in the Taylor-Couette geometry and presents preliminary tomographic PIV measurements of the growth of turbulent spots from finite-amplitude perturbations. The main results are summarized in Chapter 7.