In the absence of a rigorous and complete theory of quantum gravity, semi-classical gravity is still at the limit of our knowledge. It is well-known, however, that semi-classical theories do not neecessarily define equations of standard hyperbolic form. This is in particular the case of semi-classical gravity. In this talk, I review and give perspectives on how to obtain solutions in semi-classical theories. The perspectives come from analysing in detail a simple model (without gauge) of two weakly interacting scalar fields, one of which is quantum and sources, through its (renormalised vacuum polarisation) expectation value, the classical one. Meanwhile, the classical field is a potential in the defining quantum field equation. The point will be to obtain solutions as corrections to a leading hyperbolic system, which is well posed. Finally, I will discuss the effect of state reduction in semi-classical theories.