Hyperspectral video imaging remains a challenging task given the high dimensionality of the datasets and the limited imaging spatio-spectral-temporal tradeoffs via current methods. Yet, it has great potential in studying a variety of dynamic optical phenomena, both in microscopic and macroscopic systems. The first part of this thesis describes the design and implementation of spatially compressive hyperspectral imaging for dark-field and broad-band sum-frequency generation microscopy in order to capture and analyze different nanomaterial properties. Next, a compressive classification method using secant patterns is designed to perform task-aware compressive sensing. It achieves fast and efficient classification based on sampling but not full reconstruction using single-pixel camera hardware. Lastly, a novel compressive imaging system, the single-doxel imager (SDI), is demonstrated for four dimensional hyperspectral video imaging. It is uniquely based on a single light modulator and a single detector. By performing optical spatial and spectral modulations simultaneously with a set of designed spatio-spectral modulation patterns, it can encode hyperspectral information into a highly compressed sequence of measurements. Along with the novel optical design, a new compressive imaging reconstruction algorithm is also implemented, which is able to exploit the inherent redundancy in the 4D temporal-spatio-spectral datacube. Using this system, single-pixel hyperspectral video imaging that achieves a compression ratio of 900 to 1 is demonstrated.