Mixed noise pollution in HSI severely disturbs subsequent interpretations and applications. In this technical review, we first give the noise analysis in different noisy HSIs and conclude crucial points for programming HSI denoising algorithms. Then, a general HSI restoration model is formulated for optimization. Later, we comprehensively review existing HSI denoising methods, from model-driven strategy (nonlocal mean, total variation, sparse representation, low-rank matrix approximation, and low-rank tensor factorization), data-driven strategy 2-D convolutional neural network (CNN), 3-D CNN, hybrid, and unsupervised networks, to model-data-driven strategy. The advantages and disadvantages of each strategy for HSI denoising are summarized and contrasted. Behind this, we present an evaluation of the HSI denoising methods for various noisy HSIs in simulated and real experiments. The classification results of denoised HSIs and execution efficiency are depicted through these HSI denoising methods. Finally, prospects of future HSI denoising methods are listed in this technical review to guide the ongoing road for HSI denoising. The HSI denoising dataset could be found at https://qzhang95.github.io.