This is an expanded version of the notes of our lectures given at the conference Current Developments in Mathematics 2003 held at Harvard University on November 21-22, 2003. We present an overview of the main definitions, results and applications of the theory of cluster algebras.

I shall emphasize the role of automorphic forms in harmonic analysis because they are indispensable in analytic number theory, which is my primary subject of interest. A lot has been presented to the general forum about modular forms in algebraic number theory, in particular after resolution of the Fermat last theorem [W], [BCDT]. Therefore I shall limit my venture to the topics which help to grasp the essence of modularity in number theory as a whole.

The goal of this survey paper is to present results on hyperbolicity of complex algebraic manifolds, which appeared after the papers [48] and [42], where a number of important and intriguing conjectures were proposed. Since the paper puts the accent on measure hyperbolicity on one hand, and on algebraic methods on the other hands, we hope it will not overlap too much with the beautiful paper [21] by Demailly, except for the basic definitions and starting points. The basic questions asked in [42], and in a different spirit in [48] concern the relationships between curvature properties of a given complex manifold (or complex algebraic variety) on one hand, and the behaviour of holomorphic maps from disks or polydisks to them, on the other hand, or, from a more algebraic point of view, the (non) existence of subvarieties of certain types.