Moments are widely used in pattern recognition, image processing and computer vision. To clarify and to guide the use of different types of moments, we present, in this paper, a comparison study of the behavior of different moments. After a brief introduction to geometric, Legendre, Hermite and Gaussian-Hermite moments, we analyze at first their behavior in spatial domain. Our analysis shows orthogonal moment base functions of different orders have different number of zero crossings and very different shapes, therefore they can better separate image features based in different nodes, which is very interesting for pattern analysis and shape classification. Moreover, Gaussian-Hermite moment base functions are much more smoothed, they are thus less sensitive to noise and avoid the artifacts introduced by window function discontinuity. We then analyze the spectral behavior of moments in frequency domain. Theoretical and numerical analyses show that Legendre and Gaussian-Hermite moments of different orders separate different frequency bands more effectively. It is also shown that Gaussian-Hermite moments give an approach to construct orthogonal features from wavelet analysis results. The orthogonality equivalence theorem is also presented. Our analysis is confirmed by numerical results, which are reported also.