This dissertation is a summary of my investigations of the effects of spinpolarized currents on the dynamics of nanomagnets in which the magnetization has a vortex configuration. The "active" region of the devices consists of two ferromagnetic layers (typically made of Ni81 Fe19 ) separated by a nonmagnetic "spacer" made of Cu. The devices are lithographically patterned into nanoscale pillar structures. To obtain a stable magnetic vortex one of the two ferromagnetic layers is considerably thicker (typically 60 nm) than the exchange length of Ni81 Fe19 (~ 5 nm), making the vortex configuration more energetically favorable than the single-domain. Transfer of angular momentum from a spin-polarized current to a ferromagnet provides an efficient means to control the dynamics of nanomagnets, and consequently has been one of the most active areas of research in the field of magnetism over the past decade, driven in part by the potential for applications such as non-volatile magnetic memories and tunable, dc-driven gigahertzfrequency oscillators. Prior to my work, spin-torque driven oscillations of the magnetization had been investigated primarily in devices containing spatially uniform nanomagnets. In contrast, my experiments demonstrate that a dc spin-polarized current can be used to drive steady-state oscillations of a magnetic vortex in a spin- valve nanopillar. Detection of these oscillations is accomplished by measuring the time-varying voltage generated via the giant magneto-resistance effect. I investigated the decoherence mechanisms in these oscillators through a combination of frequency-domain and single-shot time-domain measurements. I found that, surprisingly, vortex oscillations can exhibit considerably narrower linewidths than uniform oscillations, which means that they can be a more coherent source of microwaves than vortex-free spin-torque oscillators. Yet the vortex oscillation modes also exhibit a substructure characterized by slow, discrete fluctuations that provides important insight into the possible sources of decoherence. In addition to electronic transport measurements I have also used circularlypolarized x-rays to obtain the first time-resolved, real-space images of a spintorque oscillator. These images show that the vortex has an unexpectedly complex magnetization profile resulting from the characteristically small size of these devices, and suggest that this complexity plays an important role for the excitation of steady-state vortex oscillations.