Motived by the notion of the algebraic hyperbolicity introduced by X. Chen, we introduce the notion of Nevanlinna hyperbolicity for a pair (X, D), where X is a projective variety and D is an effective Catier divisor on X. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by Paun and Sibony. This is a joint work with Yan He.