Baxter permutations are permutations whose matrix representations correspond to (renormalized) configurations of points given by the intersections (in the unit square $[0,1]^2$) of two plotted functions $y=f(x)$ and $x=g(y)$. We will review on their rich combinatorial properties and bijective connections, and give some open questions regarding the asymptotic behaviour of random Baxter permutations.