This paper introduces two fast digital implementations of the second generation curvelet transform in two and three dimensions. The first implementation uses unequally spaced fast Fourier transforms (USFFT), while the second uses a wrapping approach. Both methods are efficient, operating in O(n² log n) flops for n x n arrays and are invertible with similar complexity. They provide more accurate and efficient curvelet transforms compared to earlier methods, which were less efficient and more redundant. The software CurveLab, which implements these transforms, is available online. The curvelet transform is a multiscale geometric transform that excels at representing objects with curved or sheet-like features, offering improved sparsity and efficiency in tasks like image reconstruction and signal processing. The paper details the mathematical foundations of the curvelet transform, its digital implementations, and their applications in various scientific and engineering fields. The two digital transforms differ in how they translate curvelets at each scale and angle, with the USFFT-based method using a tilted grid and the wrapping method using a regular grid. Both methods are designed to be fast, accurate, and suitable for digital data processing. The paper also discusses the computational complexity, invertibility, and practical applications of these transforms.This paper introduces two fast digital implementations of the second generation curvelet transform in two and three dimensions. The first implementation uses unequally spaced fast Fourier transforms (USFFT), while the second uses a wrapping approach. Both methods are efficient, operating in O(n² log n) flops for n x n arrays and are invertible with similar complexity. They provide more accurate and efficient curvelet transforms compared to earlier methods, which were less efficient and more redundant. The software CurveLab, which implements these transforms, is available online. The curvelet transform is a multiscale geometric transform that excels at representing objects with curved or sheet-like features, offering improved sparsity and efficiency in tasks like image reconstruction and signal processing. The paper details the mathematical foundations of the curvelet transform, its digital implementations, and their applications in various scientific and engineering fields. The two digital transforms differ in how they translate curvelets at each scale and angle, with the USFFT-based method using a tilted grid and the wrapping method using a regular grid. Both methods are designed to be fast, accurate, and suitable for digital data processing. The paper also discusses the computational complexity, invertibility, and practical applications of these transforms.