This thesis presents theoretical groundwork for the investigation of particular phenomena of instability and decay in out-of-equilibrium many body systems. By employing mathematical methods of integrability, we succeed in gaining new insights into problems not traditionally associated with an integrable realm. The family of homogeneous sine-Gordon models provides a rare example of an integrable quantum field theory where both stable and unstable excitations are present in the spectrum. We studied the effect of unstable particles in different out-of-equilibrium settings and have shown that their presence is associated with specific signatures in many quantities of physical interest. Adopting the generalized hydrodynamic approach, we investigated the non-equilibrium dynamics following a quench and observed hallmarks of the creation and decay of unstable particles. Furthermore, we explored in considerable detail their quantitative and qualitative dependence on the parameters of the problem. Decay processes due to environmental interactions are an experimentally relevant problem, yet the theory of open systems is substantially less developed than for their closed counterparts. We concentrate on Markovian open quantum systems described by a Lindblad time evolution equation and design a framework for analytic investigation of Lindblad dynamics via establishing a connection with integrability. We identify a number of Lindblad equations that can be mapped onto non-Hermitian interacting Yang-Baxter integrable models. The key ingredient of the proposed map is the structural similarity between the operators considered. We present examples of mappings between integrable Hamiltonians and Lindblad generators and underline several models of physical interest which can be investigated within this framework. By employing Bethe ansatz techniques, we study the dynamics of some of these models.