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Gibbs phenomenon appears in numerous scientific problems and applications, for instance, the "edge effect" in image processing, or the "echo" in audio filtering. It also occurs in many numerical procedures such as the "Gibbs disaster" in nonlinear difference schemes. In the case of a function or a signal having jump discontinuity, the phenomenon is present whenever truncated orthogonal or non-orthogonal series expansions are used in approximation theory. Without exception, Gibbs phenomenon also appears in the wavelet expansion theory. In this talk, we will give a brief review of the history of this phenomenon and present recent approaches to deal with it.

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