The main subject of interest in this thesis is the existence of extremal metrics. Let (M, J, g) be a compact Kähler manifold. An extremal metric on M is a Kähler metricwhose L2 norm of the scalar curvature is minimal amongst the metrics representing the same Kähler class. New constructions of extremal metrics are explained using perturbative methods. In the first part, it is shown that if (M, J, g) is an orbisurface with isolated singularities of Hirzebruch-Jung type, then a minimal resolution of (M, J) admits extremal metrics. As an application of this result, new examples of extremal metrics are built on blow-ups of parabolic ruled surfaces. In the second part of the thesis, the stability of extremal metrics under complex deformations is studied. This is a joint work with Y.Rollin and S.Simanca. A sufficient criterion is given to ensure the existence of extremal metrics on deformations of extremal manifolds endowed with a holomorphic group action. It gives generalisation of stability results of C.Lebrun and S.Simanca. It also recovers a result of Donalsdon, that is the existence of Käler-Einstein metrics on a special deformations of the Mukai-Umemura 3-fold.
