Estimating the arrival time of a seismic wavefront is critical for subsurface imaging and earthquake localization. Traditionally, approximations of travel time were made by solving a set of factored wave equations (known as the eikonal equations) using finite-difference or finite-element methods, with events localized from these travel times by iterative optimization. Recently however, there has been growing interest in solving these problems using neural networks, given the efficiency and flexibility of such methods. Single-layer and multi-layered neural networks were trained, evaluated, and compared in their ability to approximate arrival time fields and event hypocenters in 3-D complex media. Progress toward neural networks that can accurately and reliably predict both arrival time fields and event location is shown, the strengths and weaknesses of the approach discussed, and the role that such models may play in tomographic and joint inversions is explored.