  These notes grew out of a mini-course given from May 13th to May 17th at UQAM
in Montreal during a workshop on Diophantine Approximation and Value
Distribution Theory. We start with an overview of Lang-Vojta's conjectures on
pseudo-hyperbolic projective varieties. These conjectures relate various
different notions of hyperbolicity. We begin with Brody hyperbolicity and
discuss conjecturally related notions of hyperbolicity in arithmetic geometry
and algebraic geometry in subsequent sections. We slowly work our way towards
the most general version of Lang-Vojta's conjectures and provide a summary of
all the conjectures, as well as a survey of recent progress.
