  In this thesis we study the principle that extremal objects in differential
geometry correspond to stable objects in algebraic geometry. In our
introduction we survey the most famous instances of this principle with a view
towards the results and background needed in the later chapters. In Part I we
discuss the notion of a $Z$-critical metric recently introduced in joint work
with Ruadha\'i Dervan and Lars Martin Sektnan. We prove a correspondence for
existence with an analogue of Bridgeland stability in the large volume limit,
and study important properties of the subsolution condition away from this
limit, including identifying the analogues of the Donaldson and Yang-Mills
functionals for the equation. In Part II we study the recent theory of optimal
symplectic connections on K\"ahler fibrations in the isotrivial case. We prove
a correspondence with the existence of Hermite-Einstein metrics on holomorphic
principal bundles.
