Large-scale linear operations are the cornerstone for performing complex computational tasks. Using optical computing to perform linear transformations offers potential advantages in terms of speed, parallelism, and scalability. Previously, the design of successive spatially engineered diffractive surfaces forming an optical network was demonstrated to perform statistical inference and compute an arbitrary complex-valued linear transformation using narrowband illumination. We report deep-learning-based design of a massively parallel broadband diffractive neural network for all-optically performing a large group of arbitrarily selected, complex-valued linear transformations between an input and output field of view, each with N_i and N_o pixels, respectively. This broadband diffractive processor is composed of N_w wavelength channels, each of which is uniquely assigned to a distinct target transformation; a large set of arbitrarily selected linear transformations can be individually performed through the same diffractive network at different illumination wavelengths, either simultaneously or sequentially (wavelength scanning). We demonstrate that such a broadband diffractive network, regardless of its material dispersion, can successfully approximate N_w unique complex-valued linear transforms with a negligible error when the number of diffractive neurons (N) in its design is ≥2N_wN_iN_o. We further report that the spectral multiplexing capability can be increased by increasing N; our numerical analyses confirm these conclusions for N_w > 180 and indicate that it can further increase to N_w ∼ 2000, depending on the upper bound of the approximation error. Massively parallel, wavelength-multiplexed diffractive networks will be useful for designing high-throughput intelligent machine-vision systems and hyperspectral processors that can perform statistical inference and analyze objects/scenes with unique spectral properties.