The rotation invariance of the classical disc-based moments, such as Zernike moments (ZMs), pseudo-ZMs (PZMs), and orthogonal Fourier-Mellin moments (OFMMs), makes them attractive as descriptors for the purpose of recognition tasks. However, less work has been performed for the generalization of these moment functions. In this paper, four general forms are developed to obtain a class of disc-based generalized radial polynomials that are orthogonal over the unit circle. These radial polynomials are scaled to ensure numerical stability, and some useful properties are discussed for potential applications they could be used in. Then, these scaled radial polynomials are used as kernel functions to construct a series of unit discbased generalized orthogonal moments (DGMs). The variation of parameters in DGMs can form various types of orthogonal moments: 1) generalized ZMs; 2) generalized PZMs; and 3) generalized OFMMs. The classical ZMs, PZMs, and OFMMs correspond to a special case of these three generalized moments for which the free parameter α = 0. Each member of this family will share some excellent properties for image representation and recognition tasks, such as orthogonality and rotation invariance. In addition, we have also developed two algorithms, the so-called m-recursive and n-recursive methods for the computation of these proposed radial polynomials to improve the numerical stability. Experimental results show that the proposed methods are superior to the classical disc-based moments in terms of image representation capability and classification accuracy.