Moments and moment invariants are effective feature descriptors. They have widespread applications in the field of image processing. The recent researches show that fractional-order moments have notable image representation ability. Hermite polynomials are defined over the interval from negative infinity to positive one. Such unboundedness prevents us from developing fractional-order Gaussian-Hermite moments via the existing ideas or approaches. In this paper, we propose fractional-order Gaussian-Hermite moments by forcing the definition domain of Hermite polynomials to be a bounded interval, meanwhile, resorting to a value-decreasing standard deviation to maintain the orthogonality. Moreover, we successfully develop contrast, translation and rotation invariants from the proposed moments based on the inherent properties of Hermite polynomials. The reconstructions of different types of images demonstrate that the proposed moments have more superior image representation ability to the most existing popular orthogonal moments. Besides, the salient performance in invariant image recognition, noise robustness and region-of-interest feature extraction reflect that these moments and their invariants possess the stronger discrimination power and the better noise robustness in comparison with the existing orthogonal moments. Furthermore, both complexity analysis and time consumption indicate that the proposed moments and their invariants are easy to implement, they are suitable for practical engineering applications.