In the first part of this thesis we investigate the deformation theory of compact constant scalar curvature Kähler (cscK) and extremal Kähler manifolds. In particular, we will characterise deformations of cscK- and extremal Kähler manifolds which again admit a cscK- or extremal Kähler metric, in terms of stability in the sense of geometric invariant theory. In a second part we will investigate a different problem. We will extend the adiabatic methods used by Hong and Fine in the study of cscK-metrics to the study of extremal metrics on ruled manifolds. The analysis in the case of extremal metrics being more involved will bring in some new aspects in carrying out this construction.