Networks of coupled phase oscillators have been used to model a wide array of applications including circadian rhythms, flashing fireflies, Josephson junction arrays, and high-voltage electric grids. In many of these applications, synchronization is a behavior that emerges over time and is of interest to understand and optimize for. This thesis uses tools from both pure and applied mathematics to address some of the key challenges in network theory as they relate to synchronization in networks of coupled phase oscillators via two lines of research. In the first line of research, we propose a mathematical model for designing robust networks of coupled phase oscillators by leveraging a vulnerability measure that quantifies the impact of a small perturbation at an individual phase oscillator’s natural frequency on the system’s global synchronized frequencies. We apply this mathematical framework to design high-voltage electric grids that are robust to the integration of renewable energy. In the second line of research, we use graphon theory to outline conditions under which synchronization emerges on large random networks of coupled phase oscillators. In particular, we consider the Kuramoto model on Erdős-Rényi networks and show that the model will phase synchronize with high probability as the size of the network, n, tends to infinity, as long as the edge probability is asymptotically larger than the connectivity threshold, log n/n. We also consider the Sakaguchi-Kuramoto model on Erdős–Rényi networks and show that this model will frequency synchronize with high probability as n tends to infinity, as long as the edge probability is a constant greater than zero.