Work within this thesis advances optimal control algorithms for application to magnetic resonance systems. Specifically, presenting a quadratically convergent version of the gradient ascent pulse engineering method. The work is formulated in a superoperator representation of the Liouville-von Neumann equation. A Newton-grape method is developed using efficient calculation of analytical second directional derivatives. The method is developed to scale with the same complexity as methods that use only first directional derivatives. Algorithms to ensure a well-conditioned and positive definite matrix of second directional derivatives are used so the sufficient conditions of gradient-based numerical optimisation are met. A number of applications of optimal control in magnetic resonance are investigated: solid-state nuclear magnetic resonance, magnetisation-to-singlet pulses, and electron spin resonance experiments.