There is a long and fascinating history of connections between groups and polytopes. In this talk we will mostly focus on the following simple construction. Given a permutation group $G$ as a subgroup of the set of $n$ by $n$ permutation matrices, its associated permutation polytope $P(G)$ is the convex hull of the elements in $G$. For instance, the convex hull of all $n$ by $n$ permutation matrices is the permutation polytope of the symmetric group $S_n$, called the Birkhoff polytope, which is of significance in optimization and statistics. In this way, permutation groups give rise to polytopes with interesting properties, while conversely experimental observations about these geometric objects lead to non-trivial problems for permutation groups. In this non-technical survey on work by Baumeister, Friese, Gr\"{u}ninger, Guralnick, Haase, Ladisch, Perkinson, et al. we will ask (and partially answer) some natural questions: What can we say about the structure of permutation polytopes? When are they isomorphic and in what sense? What conclusions about a permutation group can we make from its associated polytope?
