The main advantage of CS is that compression takes place during the sampling phase, making possible signi can’t savings in terms of the ADC, data storage memory, down-link bandwidth, and electrical power absorption. In this context, CS can be thought as a natural candidate to optimize the capturing of Hyperspectral Images. The main objective of CS is not to perform compression; rather, CS aims at avoiding altogether the acquisition of a very large number of samples, thereby allowing to design sensors that are more effective at acquiring the signal of interest. By realizing the importance of exploiting the correlations in all three dimensions of the hyperspectral datacube, many efforts have been devoted to the design and development of reconstruction algorithms for hyperspectral imagery, but very few of them have been based on the use of CS principles in order to reduce the amount of data acquired and to lower the energy consumption of on-boards sensors of satellite. This research work has addressed all these aspects developing innovative algorithms that provide solutions to these speci c issues. In particular, we have explored the ways the Compressed Sensing technology could be extended to iterative predictive CS reconstruction algorithms to help increase the efficiency of hyperspectral data collection and storage while fully taking advantage of sparsity structure present in all three dimensions of the HSI and keeping the computational complexity at the recovery stage at a very low level. In a nutshell, this thesis has been centered around efficient iterative reconstruction mechanisms coupled with the CS framework to achieve both of the latter points.