Geostrophic turbulent eddies are ubiquitous in Earth's oceans, but they are particularly active and important in the Southern Ocean where they mix properties such as heat, salinity, and chemical/biological tracers, thus controlling a key conduit between the surface and the deep ocean. This dissertation aims to improve our understanding of these eddies, and how they affect the responses of both the Southern Ocean and global ocean circulations to changes in Southern Ocean surface wind stress. Geostrophic eddies are usually too small for ocean general circulation models (GCM) to resolve, and therefore have to be parameterized. We first study the parameterization problem in barotropic beta-plane turbulence with a quadratic bottom drag, which is arguably the most simplified yet relevant 2D model for the turbulence but has remained unexplored thus far. We propose a prognostic theory for the eddy diffusivity, a quantity that is central to eddy parameterizations. The theory matches well with a high resolution numerical model that fully resolves the eddies, and highlights the role of Rossby waves in suppressing turbulent mixing by their relative motions to the background mean flow. Second, we test state-of-the-art eddy parameterizations in several idealized channel models for the Southern Ocean, where high resolution simulations can be performed to compare to coarse resolution simulations with eddy parameterizations. We analyze the equilibrium response of the Southern Ocean circulation to changes in surface wind stress, and how bottom topography modulates this response. We find that topography significantly suppresses the Southern Ocean response through creating standing meanders, which amplify the turbulent mixing. The coarse resolution simulations with eddy parameterizations reproduce this suppression reasonably well. Third, we explore the equilibrium response of the global ocean circulation to Southern Ocean wind stress changes in an inter-hemispheric model, which builds upon the channel model by including a basin to the north, thus allowing us to explicitly model the interactions between the Southern Ocean and the basin. By comparing with a widely used theory for the global pycnocline depth, we show that the theory underestimates the equilibrium pycnocline depth while overestimating its sensitivity to wind stress changes, due to the theory's inaccurate treatment of the low latitude upwelling. Last, we use the inter-hemispheric model to investigate the time-dependent response of the global ocean circulation to a sudden change in the Southern Ocean surface wind stress. We find that the pycnocline depth and the Antarctic Circumpolar Current (ACC) need multiple millennia to fully equilibrate, while the meridional overturning circulation (MOC) adjusts within several decades to a few centuries. Theoretical relations between the pycnocline depth and the ACC & MOC, which are accurate in the equilibrium state, break down throughout the adjustment processes. The relations break down because the responses of various circulation components depend on the ocean stratification at different depths, which adjusts on distinct timescales. This break-down explains why previous theories for the time-dependent response of global pycnocline depth have been unable to capture the multi-centennial adjustment timescale.