Tha aim of this paper is to prove a boundedness theorem for families of abelian varieties with real multiplication. More generally, the authors study curves in Hilbert modular varieties from the point of view of the Green-Griffiths-Lang conjecture claiming that entire curves in complex projective varieties of general type should be contained in a proper subvariety. Using the theory of holomorphic foliations, they establish a second main theorem following Nevanlinna theory. Finally, with a metric approach, they establish the strong Green-Griffiths-Lang conjecture for Hilbert modular varieties up to finitely many possible exceptions. This paper is organized as follows: The first section is an introduction to the subject. Section 2 concerns a tangency formula. Section 3 is devoted to leafwise hyperbolicity of Hilbert modular foliations and Section 4 to Vojta’s conjecture. Section 5 concerns the second main theorem for entire curves into Hilbert modular varieties. Section 6 concerns a metric approach, where the authors follow the approach initiated in [E. Rousseau, Ann. Sci. Éc. Norm. Supér. (4) 49, No. 1, 249–255 (2016; Zbl 1361.32032)], constructing a pseudo-metric on Hilbert modular varieties. The main difference here is that the authors allow elliptic fixed points.