  The relativistic theory of structure formation in cosmology is based mainly
on linear perturbations about a homogeneous background. But we are now driven
to understand the theory of higher-order perturbations in full detail, both
from observational and theoretical points of view. An important aspect of this
lies in defining gauge-invariant perturbations at any order. We present a new
covariant approach to this based on a local separation of tensors into their
scalar, vector and tensor parts. Such a local decomposition necessarily
requires a trio of mutually annihilating differential operators which form the
basis for defining gauge-invariant objects. It makes no use of non-local
Green's functions or pre-defined gauges, and can be used to define families of
scalar, vector and tensor modes at any order one chooses.
