I felt I could not do justice to and give a proper survey on the following topics, which I'll just mention with a few references. I may have forgotten a few important ones and apologise to any reader whose favourite topic is not mentioned.
(a) Moduli of curves and abelian varieties. The work of Parshin, Zahrin and Szpiro led to the proof that there are only finitely many curves of fixed genus over a fixed function field with a prescribed set of places of bad reduction and there are only finitely many abelian varieties over a fixed function field of characteristic p, prime-to-p isogenous to a given abelian variety. See [Sz] and [MB].
(b) Varieties of general type. Lang conjectured that, on a variety of
general type over a number field, the set of rational points is not Zariski
dense.
One may try to transpose Lang's conjectures to the case of positive
characteristic, but they are trivially false already in the case of
curves. Two natural approaches to restore the conjecture, which work
for curves, are either to insist on non - isotriviality of the variety
or to look at points which are not in the image of the Frobenius map.
Unfortunately both these approaches fail already for surfaces. In fact,
there are unirational surfaces of general type in positive characteristic,
and even non-constant families of those, which provide counterexamples
to such conjectures. (See [AV2]).One may try to look at varieties with non-zero
Kodaira - Spencer class, but there seem to be counterexamples here as well.
Again the problem is due to unirational varieties.
In all these examples the surfaces have a large set of
birational endomorphisms (coming either from the Frobenius or from birational
endomorphisms of 
), and one may try to take these into account in
stating a Lang type conjecture. A rather drastic approach
is to look only at varieties which are not covered by non-general type
varieties,
but this would be an unsatisfactory and almost unverifiable conjecture due to
the fact that it not known how to tell whether a variety of general type
can be covered by a variety which is not of general type.
In the positive direction, Martin-Deschamps and Lewin-Ménégaux [MDLM] proved that there are only finitely many separable dominating rational maps from X to Y, if X,Y are given varieties with Y of general type.
An example pertaining to this is the following. Let 
be an abelian
variety and f a rational function on A. Consider the cover X of A
defined by 
. If A is defined over 
then 
is Zariski
dense in X if and only if 
for some 
,
whereas if A is not defined over 
this happens if and only if
for some 
, where 
is the Verschiebung. A proof in the case of elliptic curves is given in
[V3] and it readily generalizes to higher dimensions.
The condition on f for 
to be Zariski dense above is equivalent to
vanishing of the Kodaira-Spencer class of X. Note that if A is
simple of dimension at least 2 and f is not a p-th power, then X is of
general type.
(c) The Birch and Swinnerton-Dyer conjecture. If 
is an abelian
variety, one defines an analytic function 
and the conjecture of
Birch and Swinnerton-Dyer states that the order of vanishing of 
at
s=1 is the rank of 
and gives a formula for the leading coefficient
of the Taylor expansion of 
around s=1. Tate showed, for elliptic
curves, that the first statement implies the second up to a power of p,
which was removed by Milne, and that the conjecture was equivalent to the
finiteness of the Tate-Shafarevich group. This has been generalized to
higher dimensions, see [Mi].
(d) Existence of solutions to equations and inequalities
We concentrated so far on finiteness statements, but one also expects
that varieties which are "very rational" to have many rational points.
For example, we have the Lang-Tsen
theorem that if 
are homogeneous
polynomials over K of degrees 
in at least 
variables, then they have a common non-trivial zero. Carlitz generalized
Tsen's method to deal with solutions of diophantine "inequalities" too.
See [Ca], [Gre].