Orthogonal moments were successfully used to extract features from gray-scale and color images. Recently, scientists show that orthogonal moments of fractional-orders have better capabilities to extract the fine features. In this work, novel orthogonal generic fractional-order Jacobi-Fourier moments are proposed for image processing, pattern recognition and computer vision applications. Novel orthogonal JacobiFourier polynomials of fractional-order were derived and defined in polar coordinates. The mathematical equation for orthogonality was formulated and a three-term recurrence relation was derived for easier computation of these polynomials. The proposed orthogonal fractional-order Jacobi-Fourier moments are generic where other orthogonal fractional-order moments are derived as special cases by choosing different values of the controlling parameters. The invariance to geometric transformations, rotation, scaling and translations, is proved where the required mathematical formulae for these invariances are presented. The proposed new fractional-order moments were tested using different datasets of gray-scale and color images in terms of image reconstruction, invariance to geometric transformations, robustness to noise, image recognition, and computational times where their performance were compared with the recent existing orthogonal integer- and fractional-order moments. The proposed generic fractional-order JacobiFourier moments outperformed all existing orthogonal moments.