The article gives an exposition of several important results obtained since the beginning of the 90's, concerning the existence of global sections of adjoint linear systems. The main goal is a deepened investigation of the structure of projective varieties. A strong motivation of this study has been the conjecture asserted by T. Fujita in 1987 : if $L\to X$ is an ample line bundle, $|K_X+mL|$ is base point free for $m\ge \dim X+1$ and very ample for $m\ge \dim X+2$. The presentation is centered around the analytic aspects of the theory : $L^2$ estimates for $\overline \partial $ operators, singular hermitian metrics, Nadel multiplier ideal sheaves. As an application, we give an overview of the proof of the conjecture of invariance of plurigenera in the case of varieties of general type, as it has been recently announced by Y.T. Siu.