  The Racah algebra and its higher rank extension are the algebras underlying
the univariate and multivariate Racah polynomials. In this paper we develop two
new models in which the Racah algebra naturally arises as symmetry algebra,
namely the Bargmann model and the Barut-Girardello model. We show how both
models are connected with the superintegrable model of Miller et al. The
Bargmann model moreover leads to a new realization of the Racah algebra of rank
$n$ as $n$-variable differential operators. Our conceptual approach also allows
us to rederive the basis functions of the superintegrable model without
resorting to separation of variables.
