This paper introduces spherical convolutional neural networks (CNNs) for analyzing spherical signals. Traditional CNNs are effective for 2D planar images but fail for spherical data due to distortions introduced by planar projections. Spherical CNNs use rotation-equivariant operations to analyze signals on the sphere, avoiding these distortions. The key idea is to define a spherical cross-correlation that is both expressive and rotation-equivariant. This operation satisfies a generalized Fourier theorem, allowing efficient computation using a generalized non-commutative Fast Fourier Transform (FFT). The paper demonstrates the computational efficiency, numerical accuracy, and effectiveness of spherical CNNs in tasks such as 3D model recognition and molecular energy regression. The spherical CNNs are built using a generalized Fourier transform for the sphere $ S^2 $ and the rotation group $ SO(3) $. The spherical correlation is defined as an inner product between a rotated input signal and a filter, resulting in a function on $ SO(3) $. This approach allows for rotation-equivariant processing of spherical data. The paper also addresses the computational challenges of implementing spherical CNNs, including the need for efficient interpolation and the high computational cost of naive implementations. These challenges are overcome using techniques from non-commutative harmonic analysis. The paper evaluates the performance of spherical CNNs on several tasks, including rotation-invariant classification of spherical images, 3D shape classification, and molecular energy regression. Results show that spherical CNNs outperform traditional CNNs in rotation-invariant tasks and achieve state-of-the-art results in 3D model recognition and molecular energy prediction. The spherical CNNs are also shown to be effective in handling spherical data without excessive feature engineering or task-specific tuning. The paper concludes that spherical CNNs represent an important advancement in the field of neural networks for spherical data, with potential applications in areas such as omnidirectional vision and global weather modeling.This paper introduces spherical convolutional neural networks (CNNs) for analyzing spherical signals. Traditional CNNs are effective for 2D planar images but fail for spherical data due to distortions introduced by planar projections. Spherical CNNs use rotation-equivariant operations to analyze signals on the sphere, avoiding these distortions. The key idea is to define a spherical cross-correlation that is both expressive and rotation-equivariant. This operation satisfies a generalized Fourier theorem, allowing efficient computation using a generalized non-commutative Fast Fourier Transform (FFT). The paper demonstrates the computational efficiency, numerical accuracy, and effectiveness of spherical CNNs in tasks such as 3D model recognition and molecular energy regression. The spherical CNNs are built using a generalized Fourier transform for the sphere $ S^2 $ and the rotation group $ SO(3) $. The spherical correlation is defined as an inner product between a rotated input signal and a filter, resulting in a function on $ SO(3) $. This approach allows for rotation-equivariant processing of spherical data. The paper also addresses the computational challenges of implementing spherical CNNs, including the need for efficient interpolation and the high computational cost of naive implementations. These challenges are overcome using techniques from non-commutative harmonic analysis. The paper evaluates the performance of spherical CNNs on several tasks, including rotation-invariant classification of spherical images, 3D shape classification, and molecular energy regression. Results show that spherical CNNs outperform traditional CNNs in rotation-invariant tasks and achieve state-of-the-art results in 3D model recognition and molecular energy prediction. The spherical CNNs are also shown to be effective in handling spherical data without excessive feature engineering or task-specific tuning. The paper concludes that spherical CNNs represent an important advancement in the field of neural networks for spherical data, with potential applications in areas such as omnidirectional vision and global weather modeling.