Campana proposed a series of conjectures relating algebro-geometric and complex-analytic properties of algebraic varieties and their arithmetic. The main ingredient is the definition of the class of special varieties, which conjecturally identity the class of varieties with a potential dense set of rational points (when defined over a number field) and admitting a dense entire curve (when defined over the complex numbers). In the talk we will review the main conjectures and constructions, and we will discuss some recent results that give evidence for some of these conjectures. This is joint work with E. Rousseau and J. Wang