  The necessity of quantising the gravitational field is still subject to an
open debate. In this paper we compare the approach of quantum gravity, with
that of a fundamentally semi-classical theory of gravity, in the weak-field
non-relativistic limit. We show that, while in the former case the
Schr\"odinger equation stays linear, in the latter case one ends up with the
so-called Schr\"odinger-Newton equation, which involves a nonlinear, non-local
gravitational contribution. We further discuss that the Schr\"odinger-Newton
equation does not describe the collapse of the wave-function, although it was
initially proposed for exactly this purpose. Together with the standard
collapse postulate, fundamentally semi-classical gravity gives rise to
superluminal signalling. A consistent fundamentally semi-classical theory of
gravity can therefore only be achieved together with a suitable prescription of
the wave-function collapse. We further discuss, how collapse models avoid such
superluminal signalling and compare the nonlinearities appearing in these
models with those in the Schr\"odinger-Newton equation.
