  We study disformal transformations of the metric in the cosmological context.
We first consider the disformal transformation generated by a scalar field
$\phi$ and show that the curvature and tensor perturbations on the uniform
$\phi$ slicing, on which the scalar field is homogeneous, are non-linearly
invariant under the disformal transformation. Then we discuss the
transformation properties of the evolution equations for the curvature and
tensor perturbations at full non-linear order in the context of spatial
gradient expansion as well as at linear order. In particular, we show that the
transformation can be described in two typically different ways: one that
clearly shows the physical invariance and the other that shows an apparent
change of the causal structure. Finally we consider a new type of disformal
transformation in which a multi-component scalar field comes into play, which
we call a "multi-disformal transformation". We show that the curvature and
tensor perturbations are invariant at linear order, and also at non-linear
order provided that the system has reached the adiabatic limit.
