We present an overview of the new concept of quasi-limit cycles at the intersection of the the theories of limit cycles and quasi-periodicity. In particular we introduce how to investigate the conditions of existence, uniqueness, and stability of strictly almost and pseudo-almost limit cycles, and almost and pseudo-automorphic limit cycles, as well as the related notion of almost andpseudo isochrons of these quasi-limit cycles. We include several illustrative examples including some relevant perturbations of the Poincaré oscillator and a special class of the renowned Van der Pol systems under suitable external forcing terms. In application some quasi-periodic waves are presented. We also include some interesting open problems for future research in order to settle these new concepts, such as the co-existence of limit cycles and strictly quasi-limit cycles, the accumulation of quasi-limit cycles, the bifurcation of quasi-limit cycles in parameterized systems, the existence of isochronous quasi-limit cycles, most importantly the transition of these types of quasi-periodic behavior to chaotic behavior through coupling and synchronization of self-sustained quasi-oscillators.
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