In this talk, we present recent results on the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in ${\mathbb{R}}^n$. We show that under minimal assumptions on $X$, for $N \gtrsim n$ and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector -- namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on $X$ we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when $X$ is $q$-stable or when $X$ has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing -- noise blind sparse recovery. This is joint work with the speaker's PhD student Christian K$\ddot{\text{u}}$mmerle (now at Johns Hopkins University) as well as Olivier Gu{\'e}don (University of Paris-Est Marne La Vall{\'e}e), Shahar Mendelson (Sorbonne University Paris), and Holger Rauhut (RWTH Aachen).