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In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first higher-dimensional Hamiltonian no-torsion theorems beyond the symplectically aspherical case. We start by showing that closed symplectic Calabi-Yau and negative monotone symplectic manifolds do not admit Hamiltonian torsion. Going still beyond topological constraints, we prove that every closed positive monotone symplectic manifold $(M,ω)$ admitting Hamiltonian torsion is geometrically uniruled by holomorphic spheres for every $ω$-compatible almost complex structure, partially answering a question of McDuff-Salamon. This provides many additional no-torsion results, and as a corollary yields the geometric uniruledness of monotone Hamiltonian $S^1$-manifolds, a fact closely related to a celebrated result of McDuff from 2009. Moreover, the non-existence of Hamiltonian torsion implies the triviality of Hamiltonian actions of lattices like $SL(k,\mathbb{Z})$ for $k \geq 2,$ as well as those of compact Lie groups. Finally, for monotone symplectic manifolds admitting Hamiltonian torsion, we prove an analogue of Newman's theorem on finite transformation groups for several natural norms on the Hamiltonian group: such subgroups cannot be contained in arbitrarily small neighborhoods of the identity. Our arguments rely on generalized Morse-Bott methods, as well as on quantum Steenrod powers and Smith theory in filtered Floer homology.