Many high-concentration multiphase materials that are industrially and academically important do not reach thermodynamic equilibrium because of kinetic constraints originating from the structurally arrested microstructure. However, owing to thermal or elastic energy of the constituents, their microstructure progressively reorganizes to form thermodynamically more stable states. In this physical ageing process, the whole spectrum of relaxation times evolves that makes relaxation dynamics increasingly sluggish. In a process of rejuvenation, deformation field increases mobility, but non-trivially alters the shape of relaxation time spectrum. Interestingly, this class of materials also demonstrates yield stress, whose origin could be closely related to physical aging, and as a result, it depends on time and deformation history. In this review, we present an overview of experimental behaviors and theoretical advances that indicate a complex coupling between ageing and rejuvenation for this class of materials having diverse microstructures.