This talk will show how the applied field of Compressive Sensing offers particularly nice insights on deep results about the geometry of high- dimensional $\ell_1^N$-balls. After reviewing the main theoretical results in Compressive Sensing, we will specifically focus on three topics: the neighborliness of the images of $\ell_1^N$-balls under random projections, the Kashin decompositions of the $\ell_1^N$-space as two orthogonal almost- Euclidean subspaces, and the Gelfand widths of $\ell_1^N$-balls relative to $\ell_2^N$.