  We lay the foundations of the first-order model theory of Coxeter groups.
Firstly, with the exception of the $2$-spherical non-affine case (which we
leave open), we characterize the superstable Coxeter groups of finite rank,
which we show to be essentially the Coxeter groups of affine type. Secondly, we
characterize the Coxeter groups of finite rank which are domains, a central
assumption in the theory of algebraic geometry over groups, which in many
respects (e.g. $\lambda$-stability) reduces the model theory of a given Coxeter
system to the model theory of its associated irreducible components.
  In the second part of the paper we move to specific definability questions in
right-angled Coxeter groups (RACGs) and $2$-spherical Coxeter groups. In this
respect, firstly, we prove that RACGs of finite rank do not have proper
elementary subgroups which are Coxeter groups, and prove further that
reflection independent ones do not have proper elementary subgroups at all.
Secondly, we prove that if the monoid $Sim(W, S)$ of $S$-self-similarities of
$W$ is finitely generated, then $W$ is a prime model of its theory. Thirdly, we
prove that in reflection independent RACGs of finite rank the Coxeter elements
are type-determined. We then move to $2$-spherical Coxeter groups, proving that
if $(W, S)$ is irreducible, $2$-spherical even and not affine, then $W$ is a
prime model of its theory, and that if $W_{\Gamma}$ and $W_{\Theta}$ are as in
the previous sentence, then $W_{\Gamma}$ is elementary equivalent to
$W_{\Theta}$ if and only if $\Gamma \cong \Theta$, thus solving the elementary
equivalence problem for most of the $2$-spherical Coxeter groups.
  In the last part of the paper we focus on model theoretic applications of the
notion of reflection length from Coxeter group theory, proving in particular
that affine Coxeter groups are not connected.
