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Hyperspectral images provide detailed spectral information through hundreds of (narrow) spectral channels (also known as dimensionality or bands) with continuous spectral information that can accurately classify diverse materials of interest. The increased dimensionality of such data makes it possible to significantly improve data information content but provides a challenge to the conventional techniques (the so-called curse of dimensionality) for accurate analysis of hyperspectral images. Feature extraction, as a vibrant field of research in the hyperspectral community, evolved through decades of research to address this issue and extract informative features suitable for data representation and classification. The advances in feature extraction have been inspired by two fields of research, including the popularization of image and signal processing as well as machine (deep) learning, leading to two types of feature extraction approaches named shallow and deep techniques. This article outlines the advances in feature extraction approaches for hyperspectral imagery by providing a technical overview of the state-of-the-art techniques, providing useful entry points for researchers at different levels, including students, researchers, and senior researchers, willing to explore novel investigations on this challenging topic. In more detail, this paper provides a bird's eye view over shallow (both supervised and unsupervised) and deep feature extraction approaches specifically dedicated to the topic of hyperspectral feature extraction and its application on hyperspectral image classification. Additionally, this paper compares 15 advanced techniques with an emphasis on their methodological foundations in terms of classification accuracies. Furthermore, the codes and libraries are shared at https://github.com/BehnoodRasti/HyFTech-Hyperspectral-Shallow-Deep-Feature-Extraction-Toolbox.