In a plethora of natural phenomena, events occur in flurries, clusters, or bunches. Modern service systems are no exception to this. This can be by design, such as in batches of jobs being sent to a data center for processing, or simply by circumstance, such as in bursts of newly infected flu patients arriving to a health clinic or in the virality of new interactions with a popular social media post. This thesis is concerned with the modeling, exploration, and analysis of these batch and burst arrival processes through the lens of applied probability. Often, this builds on the idea of self-exciting Hawkes process, in which each arrival increases the likelihood of another arrival occurring soon after, forming quick bursts of several successive arrivals. By comparison, batches are taken to be truly simultaneous, with multiple entities entering the system at precisely the same epoch. In the course of this dissertation, batches are both compared to bursts and used as tools to develop deeper understanding of bursts. These objects are also both applied in a variety of settings, most notably in the problem of staffing teleoperation support systems for autonomous vehicles. This analysis reveals that batches and bursts have a pronounced effect on service systems, and thus must be addressed.