This thesis studies the limiting behaviour of the torsion in the homotopy groups πn(X) of a space X as n → ∞. It is a ‘three paper thesis’, the main body of which consists of the following papers:[1] G. Boyde, Bounding size of homotopy groups of spheres. Proceedings of the Edinburgh Mathematical Society, 63(4):1100–1105, 2020.[2] G. Boyde, p-hyperbolicity of homotopy groups via K-theory, preprint, available at arXiv:2101.04591 [math.AT], 2021.[3] G. Boyde, Z/pr-hyperbolicity via homology, preprint, available at arXiv:2106.03516 [math.AT], 2021.In [1], we improve on the best known bound for the size of the homotopy group πq(Sn), using the combinatorics of the EHP sequence.In [2], we study Huang and Wu’s p- and Z/pr -hyperbolicity for spaces related to the wedge of two spheres Sn ∨ Sm. We show that Sn v SmSn ∨ S m Sn ∨ S m Sn ∨ SmSn V SmSn ∨ Sm. Sn ∨ Sm.is Z/pr -hyperbolic for all primes p and all r ∈ N, which implies that various spaces containing Sn ∨ Sm as a retract are similarly hyperbolic. We then prove a K-theory criterion for p-hyperbolicity of a finite suspension ΣX, and deduce some examples.In [3], we study p- and Z/pr -hyperbolicity for spaces related to the Moore space Pn (pr). When ps≠2, we show that Pn (pr) is Z/ps -hyperbolic for s ≤ r. Combined with Huang and Wu’s work, and Neisendorfer’s results on homotopy exponents, this completely resolves the question of when such a Moore space is Z/ps -hyperbolic for p ≥ 5. We then prove a homological criterion for Z/pr -hyperbolicity of a space X, and deduce some examples.