The study of nonlinear dynamics, specifically chaotic dynamics, has been an active area of interdisciplinary research since the 1970s. Today, researchers are interested in practical applications of chaos, such as chaos communications and ranging, which require simple devices that produce complex and high-speed dynamics. Furthermore, since all physical signals travel at a finite speed, it is important to understand how inherent time delays in both natural and man-made systems interact with nonlinearities to affect their behavior. One of the first nonlinear time-delay systems to be thoroughly studied was an optical ring cavity proposed by Ikeda in 1979. I will present an overview of his research on the dynamics of systems with time-delayed low-pass feedback, of which the Ikeda system is one example. Next, I will discuss the properties of systems with time-delayed bandpass feedback. Finally, I will conclude with an analysis of one particular time-delayed bandpass feedback device, known as an optoelectronic oscillator. I found that this device is capable of producing high-speed chaotic behavior with a flat, broad power spectrum extending out to 8 GHz (the cutoff frequency of the oscilloscope used to measure the dynamics) as well as a train of short pulses spaced by the round-trip time of the feedback loop. This oscillator not only provides a simple device for fundamental studies of time-delay dynamical systems, but the observed broadband chaotic and pulsing behaviors of this device also make it a promising candidate for use in applications such as chaos communication, ultrawide-band sensor networks, and frequency comb generation.