  We prove that there are no unexpected universal integral linear relations and
congruences between Hodge, Betti and Chern numbers of compact complex manifolds
and determine the linear combinations of such numbers which are bimeromorphic
or topological invariants. This extends results in the K\"ahler case by
Kotschick and Schreieder. We then develop a framework to tackle the more
general questions taking into account `all' cohomological invariants (e.g. the
dimensions of the higher pages of the Fr\"olicher spectral sequence, Bott-Chern
and Aeppli cohomology). This allows us to reduce the general questions to
specific construction problems. We solve these problems in many cases. In
particular, we obtain full answers to the general questions concerning
universal relations and bimeromorphic invariants in low dimensions.
