In this thesis, the main objective is to study the presence of Gibbs phenomenon and the Gibbs constant in Fourier-Legendre series. The occurrence of The Gibbs phenomenon is a well known consequence when approximating functions with Fourier series that have points of discontinuity. Consequently, the initial focus was to examine Fourier seriesand the occurrence of Gibbs phenomenon in this context. Next, we delve into Legendrepolynomials, showing their applicability to be expressed as a Fourier series due to theirorthogonality in [−1, 1]. We then continue to explore Gibbs phenomenon for Fourier-Legendre series. The findings proceeds to confirm the existence of the Gibbs phenomenon for Fourier-Legendre series, but most notebly, the values of the error seem to convergeto the same number as for Fourier series which is the Gibbs constant.