  We prove new theorems which are higher-dimensional generalizations of the
classical theorems of Siegel on integral points on affine curves and of Picard
on holomorphic maps from $\mathbb{C}$ to affine curves. These include results
on integral points over varying number fields of bounded degree and results on
Kobayashi hyperbolicity. We give a number of new conjectures describing, from
our point of view, how we expect Siegel's and Picard's theorems to optimally
generalize to higher dimensions. In some special cases we will be able to
relate our conjectures to existing conjectures. In this respect, we are also
led to formulate a new conjecture relating the absolute discriminant and height
of an algebraic point on a projective variety over a number field.
