We define bounded generation for En-algebras in chain complexes and prove that for n ≥ 2 this property is equivalent to homological stability. Using this we prove a local-to-global principle for homological stability, which says that if an En-algebra A has homological stability (or equivalently the topological chiral homology R n A has homology stability), then so has the topological chiral homology M A of any connected non-compact manifold M . Using scanning, we reformulate the local-to-global homological stability principle so that it applies to compact manifolds. We also give several applications of our results.Here k M X denotes the charge k component of M X, B T M X is a bundle over M with fiber given by the n-fold delooping B n X of X, and Γ c k (−) denotes the space of compactly