This paper introduces translation-invariant operators on \( \mathbf{L}^2(\mathbb{R}^d) \) that are Lipschitz continuous under the action of diffeomorphisms. The authors define a scattering propagator as a path-ordered product of nonlinear and non-commuting operators, each computing the modulus of a wavelet transform. A windowed scattering transform is introduced, which is proved to be Lipschitz continuous under the action of \( \mathbf{C}^2 \) diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation-invariant. The scattering coefficients also provide representations of stationary processes, with expected values depending on high-order moments and capable of discriminating processes with the same power spectrum. The paper extends scattering operators to compact Lie groups \( G \), making them invariant under the action of \( G \). Combining scattering on \( \mathbf{L}^2(\mathbb{R}^d) \) and \( \mathbf{L}^2(SO(d)) \) results in a translation and rotation-invariant scattering on \( \mathbf{L}^2(\mathbb{R}^d) \). The paper includes numerical examples and discusses the application of scattering transforms to audio and image classification.This paper introduces translation-invariant operators on \( \mathbf{L}^2(\mathbb{R}^d) \) that are Lipschitz continuous under the action of diffeomorphisms. The authors define a scattering propagator as a path-ordered product of nonlinear and non-commuting operators, each computing the modulus of a wavelet transform. A windowed scattering transform is introduced, which is proved to be Lipschitz continuous under the action of \( \mathbf{C}^2 \) diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation-invariant. The scattering coefficients also provide representations of stationary processes, with expected values depending on high-order moments and capable of discriminating processes with the same power spectrum. The paper extends scattering operators to compact Lie groups \( G \), making them invariant under the action of \( G \). Combining scattering on \( \mathbf{L}^2(\mathbb{R}^d) \) and \( \mathbf{L}^2(SO(d)) \) results in a translation and rotation-invariant scattering on \( \mathbf{L}^2(\mathbb{R}^d) \). The paper includes numerical examples and discusses the application of scattering transforms to audio and image classification.