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 applied mathematics and scientific computation in helping overcome these obstacles. I then introduce some of the mathematical models used in studying fusion stability and refueling, and how solutions to those models may be approximated. Of particular interest in such approximation techniques is the ability of the relevant numerical methods to scale up to resolutions necessary for accurately modeling the underlying physics. To that end, I discuss some of my recent work in the development of fully implicit solution approaches for magnetic fusion modeling, presenting both numerical results and theoretical analysis demonstrating the benefit of these approaches over traditional methods.