  Let $R$ be a group and let $S$ be a subset of $R$. The Haar graph
$\mathrm{Haar}(R,S)$ of $R$ with connection set $S$ is the graph having vertex
set $R\times\{-1,1\}$, where two distinct vertices $(x,-1)$ and $(y,1)$ are
declared to be adjacent if and only if $yx^{-1}\in S$. The name Haar graph was
coined by Toma\v{z} Pisanski in one of the first investigations on this class
of graphs.
  For every $g\in R$, the mapping $\rho_g:(x,\varepsilon)\mapsto
(xg,\varepsilon)$, $\forall (x,\varepsilon)\in R\times\{-1,1\}$, is an
automorphism of $\mathrm{Haar}(R,S)$. In particular, the set
$\hat{R}:=\{\rho_g\mid g\in R\}$ is a subgroup of the automorphism group of
$\mathrm{Haar}(R,S)$ isomorphic to $R$. In the case that the automorphism group
of $\mathrm{Haar}(R,S)$ equals $\hat{R}$, the Haar graph $\mathrm{Haar}(R,S)$
is said to be a Haar graphical representation of the group $R$.
  Answering a question of Feng, Kov\'{a}cs, Wang, and Yang, we classify the
finite groups admitting a Haar graphical representation. Specifically, we show
that every finite group admits a Haar graphical representation, with abelian
groups and ten other small groups as the only exceptions.
  Our work on Haar graphs allows us to improve a 1980 result of Babai
concerning representations of groups on posets, achieving the best possible
result in this direction. An improvement to Babai's related result on
representations of groups on distributive lattices follows.
