Hyperspectral imaging provides the full spectrum at each point of the whole field-of-view, and thus is being extensively employed in remote sensing, surveillance, medical diagnostics and biological research. However, the intrinsically limited photons for each spectral band and the inevitable noise during acquisition result in complex degradation of hyperspectral images (HSIs) that adversely impacts the subsequent data analysis. Yet, it remains challenging for current HSI denoising methods to effectively address HSI datasets that are significantly contaminated by complex noise, especially in terms of spectral recovery. In this paper, we propose a spatial-spectral collaborative denoising network (SSCDN) that makes full use of spatial-spectral correlation information for HSI denoising. Through the combination of attention mechanism and specifically designed spatial-spectral collaborative attention module along with a multi-loss joint optimization strategy, the proposed model achieves superior denoising performance while well-preserving spectral and spatial features for complex degradation. Extensive experimental results on simulated and real data for remote sensing and biomedical applications demonstrate that the proposed SSCDN outperforms other state-of-the-art competitive HSI denoising methods under various noise settings, especially in terms of structural-spectral fidelity and the model robustness against noise.