Several large classes of homogeneous spaces are known to be formal---in the
sense of Rational Homotopy Theory. However, it seems that far fewer examples of
non-formal homogeneous spaces are known.
In this article we provide several construction principles and
characterisations for non-formal homogeneous spaces, which will yield a lot of
examples. This will enable us to prove that, from dimension 72 on, such a space
can be found in each dimension
