In this volume of the seminar, there are 15 review lectures on topics of current interest. The volume contains seminars on Nakajima varieties (O. Schiffmann) and on the geometry of the moduli space of K3 surfaces (C. Voisin). There is a paper on Tate-Sato conjectures by H. Carayol. The lecture by O. Debarre is devoted to pluricanonical systems on varieties and the lecture by D. Harari treats rational points of subvarieties of Abelian varieties (over a function field). The theme of algebraic independence in non-zero characteristics is discussed by F. Pellarin, ordinary differential equations with BV coefficients are studied by C. De Lellis and counting the number of solutions of polynomial equations in finite fields is described by A. Chambert-Loir. There are three lectures on differential geometry: algebraization of webs (J. V. Pereira), semi-classical measures on Riemannian manifolds (Y. Colin de Verdièr) and closed contact manifolds (V. Colin). There are also two lectures on group theory: the Kashiwara-Vergne conjecture (Ch. Torossian) and groups generated by automata (A. Zuk). A paper on the solutions of the Hamilton-Jacobi equations was prepared by J.-M.Roquejoffre and the subspace theorem in the theory of Diophantine approximation is described in the contribution by Yuri F. Bilu.