Fusion energy holds the promise of a clean, sustainable and safe energy source for the future. While research in this field has been ongoing for the last half century, much work remains before it may prove a viable source of energy. In this talk, I discuss some of the scientific and engineering challenges remaining in fusion energy, and the role of mathematics research in overcoming these obstacles. In particular, I will discuss some of the mathematical models used in studying fusion stability and refueling, how solutions to those models may be approximated, and introduce some model improvements to better simulate fusion processes.