In this work, spin transport in the XXZ model is investigated. As a quantum integrable model the XXZ spin chain contains infinitely many conserved charges. The presence of these conserved charges results in the coexistence of ballistic and diffusive spin transport. Ballistic spin transport is characterized by the spin Drude weight, which was originally derived in 1999 by Zotos. This original derivation was controversial, particularly when divergences were observed in intermediate steps. In this thesis a novel approach for deriving the spin Drude weight demonstrates that Zotos' formula follows from relatively few assumptions. Furthermore, an asymptotic analysis of this formula at infinite temperature was found to be consistent with the Prosen--Ilievski bound, obtained from direct consideration of conserved charges. At low--temperatures a more complicated analysis is carried out to determine the non--analytic temperature correction term. This non--analytic temperature correction was found to give rise to the fractal structure of the Drude weight observed at finite temperatures. The form of this low--temperature correction raises important questions about the applicability of non--linear Luttinger liquid theory to quantum integrable models. Spin transport in the non--linear response framework is also considered in this thesis, where the non--linear Drude weight (NLDW) at infinite temperatures is determined. At infinite temperatures the NLDW is found to have an even more fractal structure than the linear Drude weight due to Bethe string lengths appearing explicitly in the final result. These results suggest that the Bethe strings may be indispensable for understanding spin transport.