In this paper we study three notions of hyperbolicity for varieties admitting complex polarized variation of Hodge structures ($\mathbb{C}$-PVHS for short). In the first part we prove that if a quasi-projective manifold $U$ admits a $\mathbb{C}$-PVHS whose period map is quasi-finite, then $U$ is algebraically hyperbolic in the sense of Demailly, and that the generalized big Picard theorem holds for $U$: any holomorphic map $f:\Delta-\{0\}\to U$ from the punctured unit disk to $U$ extends to a holomorphic map of the unit disk $\Delta$ into any projective compactification of $U$... This result generalizes a recent work by Bakker-Brunebarbe-Tsimerman. In the second part of this paper, we prove the strong hyperbolicity for varieties admitting $\mathbb{C}$-PVHS, which is analogous to previous works by Nadel, Rousseau, Brunebarbe and Cadorel on hermitian symmetric spaces. (read more)