Quantum states of matter-such as solids, magnets and topological phases-typically exhibit collective excitations (for example, phonons, magnons and anyons)1. These involve the motion of many particles in the system, yet, remarkably, act like a single emergent entity-a quasiparticle. Known to be long lived at the lowest energies, quasiparticles are expected to become unstable when encountering the inevitable continuum of many-particle excited states at high energies, where decay is kinematically allowed. Although this is correct for weak interactions, we show that strong interactions generically stabilize quasiparticles by pushing them out of the continuum. This general mechanism is straightforwardly illustrated in an exactly solvable model. Using state-of-the-art numerics, we find it at work in the spin- 1/2 triangular-lattice Heisenberg antiferromagnet (TLHAF). This is surprising given the expectation of magnon decay in this paradigmatic frustrated magnet. Turning to existing experimental data, we identify the detailed phenomenology of avoided decay in the TLHAF material2 Ba3CoSb2O9, and even in liquid helium3,4,5,6,7,8, one of the earliest instances of quasiparticle decay9. Our work unifies various phenomena above the universal low-energy regime in a comprehensive description. This broadens our window of understanding of many- body excitations, and provides a new perspective for controlling and stabilizing quantum matter in the strongly interacting regime.