What are the main open problems in the theory of amenability of groups? I have read the Paterson and Runde books about amenability of groups, but I do not know what are the most intriguing questions in this area today.
A survey or a list of questions would be welcome.
 A: The Part C of Appendices of the book "Amenability of discrete groups by examples"
by Kate Juschenko, contains open problems in amenability.
A: $\DeclareMathOperator\IET{IET}$ A folklore question is the amenability of the group $\IET$ of interval exchanges transformations (= right continuous permutations of the interval $[0,1\mathclose[$ that locally coincide with translations outside a finite subset).
(Unlike Thompson's group $F$, it is also unknown whether this group contains  a non-abelian free subgroup; this question is attributed to A. Katok.)
Progress on this question: for obvious reasons, amenability of $\IET$ is equivalent to amenability of all its finitely generated subgroup. For a subgroup $G$ of $\IET$, define its rank $r(G)$ as the $\mathbf{Q}$-rank of the subgroup $\Lambda_G$ of $\mathbf{R}/\mathbf{Z}$ generated by translation lengths of elements of $G$ modulo $\mathbf{Z}$. If $G$ is finitely generated, then $r(G)<\infty$.
For instance $r(G)=0$ means that $G$ has only rational translation lengths, and is then locally finite, hence (obviously) amenable.
If $r(G)\le 1$ then $G$ is amenable: this is due to Juschenko and Monod (technically, they need $\Lambda_G$ to be cyclic and not just virtually cyclic, but basically this is what their method, not specific to $\IET$, provides). Using random walks, Juschenko, Matte Bon, Monod and de la Salle proved it for $r(G)\le 2$. The recurrence of the planar random walk is used in the proof, so $r(G)=3$ is unknown at this stage, as well as the general case.
I don't know if there's any conjecture about this question. I tend to bet on a positive answer ($\IET$ is amenable) but have no evidence enough to conjecture it.
A: Does there exist an infinite finitely presented simple amenable group?
Weaker variant: does there exist an infinite finitely presented aperiodic amenable group? Here aperiodic means: no nontrivial finite quotient.
The same problems for finitely generated groups were solved about 10 years ago. The final step was due to Juschenko and Monod, who proved the amenability of topological-full groups of Cantor minimal $\mathbf{Z}$-subshifts (2012, published 2013). Previously Matui proved that their derived subgroup is simple and finitely generated (2006). That these groups, previously introduced by Putnam / Giordano-Putnam-Skau in the 90s, are amenable, was conjectured by Grigorchuk and Medynets in 2011.
However these groups (as well as IET) are LEF and hence their finitely presented subgroups are residually finite, so there's not even a plausible candidate at this point, as far as I know.
(I'm not sure of a reference for this question: possibly it can be found in the Baumslag or Kourovka list?)
