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This paper considers the bounded travelling waves of the RLW-Burgers equation. We prove that there only exist two types of bounded travelling waves, the monotone kink waves and the oscillatory kink waves. For the oscillatory kink wave, the regularity of its maximum oscillation amplitude changing with parameters is discussed. Exact expressions of the monotone kink waves and approximate expressions of the oscillatory ones are obtained in some special cases. Furthermore, all bounded travelling waves of the RLW-Burgers equation under different parameter conditions are identified and the evolution of them is discussed to explain the corresponding physical phenomena.