  Originally in 1954 the Tutte polynomial was a bivariate polynomial associated
to a graph in order to enumerate the colorings of this graph and of its dual
graph at the same time. However the Tutte polynomial reveals more of the
internal structure of a graph, and contains even other specializations from
other sciences like the Jones polynomial in Knot theory, the partition function
of the Pott model in statistical physics, and the reliability polynomial in
network theory. In this article, we study the Tutte polynomial associated to
more general objects which are the arrangements of hyperplanes. Indeed
determining the Tutte polynomial of a graph is equivalent to determining the
Tutte polynomial of a special hyperplane arrangement called graphic
arrangement. In 2007 Ardila computed the Tutte polynomials of the hyperplane
arrangements associated to the classical Weyl groups, and the characteristic
polynomials of the Catalan arrangements. The charateristic polynomial is also a
specialization of the Tutte polynomial. In 2012 Seo computed the characteristic
polynomials of the Shi threshold arrangements, and in 2017 Song computed the
characteristic polynomials of the $\mathcal{I}_n$ arrangements in the plane and
in the space. We aim to bring a more general result by introducing a wider
class of hyperplane arrangements which is the set of symmetric hyperplane
arrangements. We compute the Tutte polynomial of a symmetric hyperplane
arrangement, and, as examples of application, we deduce the Tutte polynomials
of the Catalan, the Shi threshold, and the $\mathcal{I}_n$ arrangements.
