Recent advances in airborne and spaceborne hyperspectral imaging technology have provided end users with rich spectral, spatial, and temporal information, which make a plethora of applications for the analysis of large areas of the Earth surface feasible. However, a huge number of factors, such as high dimensions and size of the hyperspectral data, the lack of training samples, mixed pixels, light scattering mechanisms in the acquisition process, and different atmospheric and geometric distortions, make such data inherently nonlinear and complex, which poses extreme challenges for existing methodologies to effectively process and analyze the data sets. Hence, rigorous and innovative methodologies are required for hyperspectral image and signal processing and have become a center of attention for researchers worldwide. This paper offers a comprehensive tutorial/overview focusing specifically on hyperspectral data analysis, which is categorized into seven broad topics: classification, spectral unmixing, dimensionality reduction, resolution enhancement, hyperspectral image denoising and restoration, change detection, and fast computing. For each topic, we provide a synopsis of the state-of-the-art approaches and numerical results for validation and evaluation of different methodologies, followed by a discussion of future challenges and research directions.