  Quantum states of matter---such as solids, magnets and topological
phases---typically exhibit collective excitations---phonons, magnons, anyons.
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, common wisdom says that quasiparticles become unstable when
they encounter the inevitable continuum of many-particle excited states at high
energies. Whilst correct for weak interactions, we show that this is far from
the whole story: strong interactions generically stabilise 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 also in the spin-$\frac{1}{2}$ triangular lattice Heisenberg
antiferromagnet (TLHAF) near the isotropic point---this is surprising given the
common 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 material Ba$_3$CoSb$_2$O$_9$, and even in liquid
helium---one of the earliest instances of quasiparticle decay. 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 stabilising
quantum matter in the strongly-interacting regime.
