This paper introduces translation-invariant operators on $ \mathbf{L}^{2}(\mathbb{R}^{d}) $ that are Lipschitz continuous to the action of diffeomorphisms. A scattering propagator is defined as a path-ordered product of non-linear and non-commuting operators, each computing the modulus of a wavelet transform. A windowed scattering transform is a nonexpansive operator that locally integrates the scattering propagator output. It is proved to be Lipschitz continuous to $ C^{2} $ diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform which is translation invariant. Scattering coefficients provide representations of stationary processes, and are Lipschitz continuous to random deformations. Scattering operators are extended to $ \mathbf{L}^{2}(G) $, where G is a compact Lie group, and are invariant under the action of G. Combining scattering on $ \mathbf{L}^{2}(\mathbb{R}^{d}) $ and $ \mathbf{L}^{2}(SO(d)) $ defines a translation and rotation invariant scattering on $ \mathbf{L}^{2}(\mathbb{R}^{d}) $. The paper also discusses the application of scattering transforms to audio and image classification, and introduces a software package for numerical experiments. The scattering transform is shown to have similarities with the Fourier transform modulus, but is stable to the action of diffeomorphisms. The paper proves that the scattering transform is nonexpansive and preserves the $ \mathbf{L}^{2}(\mathbb{R}^{d}) $ norm. It also shows that the scattering transform is translation invariant and converges to a translation invariant scattering transform as the window size increases. The scattering transform is also shown to be invariant under the action of compact Lie groups. The paper concludes with a conjecture on the conditions for strong convergence in $ \mathbf{L}^{2}(\mathbb{R}^{d}) $.This paper introduces translation-invariant operators on $ \mathbf{L}^{2}(\mathbb{R}^{d}) $ that are Lipschitz continuous to the action of diffeomorphisms. A scattering propagator is defined as a path-ordered product of non-linear and non-commuting operators, each computing the modulus of a wavelet transform. A windowed scattering transform is a nonexpansive operator that locally integrates the scattering propagator output. It is proved to be Lipschitz continuous to $ C^{2} $ diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform which is translation invariant. Scattering coefficients provide representations of stationary processes, and are Lipschitz continuous to random deformations. Scattering operators are extended to $ \mathbf{L}^{2}(G) $, where G is a compact Lie group, and are invariant under the action of G. Combining scattering on $ \mathbf{L}^{2}(\mathbb{R}^{d}) $ and $ \mathbf{L}^{2}(SO(d)) $ defines a translation and rotation invariant scattering on $ \mathbf{L}^{2}(\mathbb{R}^{d}) $. The paper also discusses the application of scattering transforms to audio and image classification, and introduces a software package for numerical experiments. The scattering transform is shown to have similarities with the Fourier transform modulus, but is stable to the action of diffeomorphisms. The paper proves that the scattering transform is nonexpansive and preserves the $ \mathbf{L}^{2}(\mathbb{R}^{d}) $ norm. It also shows that the scattering transform is translation invariant and converges to a translation invariant scattering transform as the window size increases. The scattering transform is also shown to be invariant under the action of compact Lie groups. The paper concludes with a conjecture on the conditions for strong convergence in $ \mathbf{L}^{2}(\mathbb{R}^{d}) $.