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).