  This thesis concerns the compatibility of inhomogeneous cosmologies with our
present understanding of the universe. It is a problem of some interest to find
the class of all relativistic cosmological models which are capable of
providing a reasonable `fit' to the universe. This thesis, in some respects, is
part of this process. We consider Stephani models, which are a generalisation
of the standard Friedmann-Lemaitre-Robertson-Walker (FLRW) models, which can be
thought of as FLRW models with acceleration and pressure gradients. Thus these
models generalise the `dust' assumption of standard cosmology.
  The crucial aspect of this work is the retention of the Copernican principle
-- an assumption regarded by many as crucial to cosmology. It states that we
are not at a special location in the universe. This is a vital aspect of the
original work in this thesis: consideration of an inhomogeneous model, while
retaining the Copernican principle has, as far as the author is aware, not been
considered in detail before.
  We start by generalising the Ehlers-Geren-Sachs Theorem to identify the class
of inhomogeneous spacetimes which allow an isotropic radiation field for all
observers in the spacetime. We then investigate observational and physical
aspects of these models from all observer locations. We conclude that there
exist spacetimes which conform to present observational constraints (especially
anisotropy constraints) for any location in the spacetime, while at the same
time being significantly inhomogeneous; ie, not `almost-FLRW'.
