  A Weyl structure is usually defined by an equivalence class of pairs $({\bf
g}, \boldsymbol{\omega})$ related by Weyl transformations, which preserve the
relation $\nabla {\bf g}=\boldsymbol{\omega}\otimes{\bf g}$, where ${\bf g}$
and $\boldsymbol{\omega}$ denote the metric tensor and a 1-form field. An
equivalent way of defining such a structure is as an equivalence class of
conformally related metrics with a unique affine connection
$\Gamma_{\boldsymbol{\omega}}$, which is invariant under Weyl transformations.
In a standard Weyl structure, this unique connection is assumed to be
torsion-free and have vectorial non-metricity. This second view allows us to
present two different generalizations of standard Weyl structures. The first
one relies on conformal symmetry while allowing for a general non-metricity
tensor, and the other comes from extending the symmetry to arbitrary
(disformal) transformations of the metric.
