The total variation (TV) regularization has been widely used in various applications related to hyperspectral (HS) signal and image processing due to its potential in modeling the underlying smoothness of HS data. However, most existing TV norms usually tend to generate spatial oversmoothing or artifacts. To this end, we propose a novel <inline-formula> <tex-math notation="LaTeX">l_{0} </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">l_{1} </tex-math></inline-formula> hybrid TV (<inline-formula> <tex-math notation="LaTeX">l_{0} </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">l_{1} </tex-math></inline-formula>HTV) regularization with the applications to HS mixed noise removal and compressed sensing (CS). More specifically, <inline-formula> <tex-math notation="LaTeX">l_{0} </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">l_{1} </tex-math></inline-formula>HTV can be regarded as a globally and locally integrated TV regularizer, where the <inline-formula> <tex-math notation="LaTeX">l_{0} </tex-math></inline-formula> gradient constraint is incorporate into the <inline-formula> <tex-math notation="LaTeX">l_{1} </tex-math></inline-formula> spatial-spectral TV (<inline-formula> <tex-math notation="LaTeX">l_{1} </tex-math></inline-formula>-SSTV). <inline-formula> <tex-math notation="LaTeX">l_{1} </tex-math></inline-formula>-SSTV is capable of exploiting the local structure information across both spatial and spectral domains, while the <inline-formula> <tex-math notation="LaTeX">l_{0} </tex-math></inline-formula> gradient can promote a globally spectral-spatial smoothness by directly controlling the number of nonzero gradients of HS images. This efficient combination considers more comprehensive prior knowledge of HS images, yielding sharper edge preservation and resolving the above drawbacks of existing pure TV norms. More significantly, <inline-formula> <tex-math notation="LaTeX">l_{0} </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">l_{1} </tex-math></inline-formula>HTV can be easily injected into HS-related processing models, and an effective algorithm based on the alternating direction method of multipliers (ADMM) is developed to solve the optimization problems. Extensive experiments conducted on several HS data sets substantiate the superiority and effectiveness of the proposed method in comparison with many state-of-the-art methods.