Abstract: I will describe a new notion of stability associated to fibrations in algebraic geometry. Much like Tian-Donaldson's notion of K-stability, there is an associated notion of a "canonical metric", in the form of an "optimal symplectic connection". Our main result shows that the existence of an optimal symplectic connection implies that the fibration is semistable. There is an associated moduli problem for stable fibrations over a fixed base, and I will also explain certain aspects of this. This is joint work with Lars Sektnan.