This paper introduces two digital implementations of the second-generation curvelet transform in two and three dimensions. The first implementation is based on unequally spaced fast Fourier transforms (USFFT), while the second is based on the wrapping of selected Fourier samples. Both implementations are designed to be fast, running in \(O(n^2 \log n)\) flops for \(n \times n\) Cartesian arrays, and are invertible with rapid inversion algorithms of similar complexity. The curvelet transform is a powerful tool for representing objects with curve-punctuated smoothness, offering optimal sparsity and efficient computation. The paper details the mathematical foundations, the algorithms for both implementations, and their properties, including their isometric nature and faithful approximation of the continuous transform. The software CurveLab, which implements these transforms, is available online.This paper introduces two digital implementations of the second-generation curvelet transform in two and three dimensions. The first implementation is based on unequally spaced fast Fourier transforms (USFFT), while the second is based on the wrapping of selected Fourier samples. Both implementations are designed to be fast, running in \(O(n^2 \log n)\) flops for \(n \times n\) Cartesian arrays, and are invertible with rapid inversion algorithms of similar complexity. The curvelet transform is a powerful tool for representing objects with curve-punctuated smoothness, offering optimal sparsity and efficient computation. The paper details the mathematical foundations, the algorithms for both implementations, and their properties, including their isometric nature and faithful approximation of the continuous transform. The software CurveLab, which implements these transforms, is available online.