For curves one would expect finiteness for rational/integral points provided
the genus is large enough and the curve is non-isotrivial. In
characteristic p > 0, already the notion of genus has a twist to it.
Let C be an algebraic curve defined over a function field K
of characteristic p > 0. One can define the (absolute) genus of C by
extending the field to the algebraic closure. Another option is to define
the genus of C relative to K to be the integer 
that makes the
Riemann-Roch formula hold, that is, for any K-divisor D of C,
of sufficiently large degree, the
dimension, 
, of the K-vector space of functions of 
whose
polar divisor is bounded by D, is 
. Since K is
not perfect, the relative genus may change under inseparable extensions.
Luckily, this type of curve is actually easier to handle and it can be
shown ([V3]) that if the genus of C relative to K is different from
the (absolute) genus of C then 
is finite. We will now restrict our
discussion to curves that do not change genus so there is now no loss of
generality in assuming the curves to be smooth.
As mentioned above, Samuel proved the Mordell conjecture, to the effect that a curve of (absolute) genus at least two, which is non-isotrivial, has only finitely many rational points over a function field.
It remains to discuss integral points on affine curves. As expected,
Sketch of proof:
Let C be such a curve and 
is completion. Then there exists a cover
X of 
branched over 
only, of degree prime to the
characteristic, such that X is of genus
at least 2. If 
has genus at least two, this is the Kodaira-Parshin
construction ([L2],[Sz],), otherwise it is elementary. It also follows from the
construction that X is non-isotrivial. Integral points on C over a fixed
ring will then lift to rational points on X over a fixed field and the
theorem then follows from the Mordell conjecture.
It is worth remarking at this point that the projective line minus three points is always isotrivial.
Another topic is to find bounds for the height of rational points on curves of genus at least 2 (i.e. effective Mordell). Szpiro ([Sz]) had the first result on this line, which was improved by Moriwaki ([Mo]) and then by Kim ([Ki]) who obtained the following result:
The bound 
is, in general,
best possible. An example showing this can be constructed as
follows.
Let 
be a curve defined over the finite field 
with q elements,
let F be
the Frobenius map of 
, 
the function field of 
and 
the generic point. Let X be a cover of 
ramified only at P if the genus
of 
is at least 2 and ramified at P and 0 if 
is
an elliptic curve.
To construct X one can use the Kodaira-Parshin construction. Now consider the
points 
, say, of X which lift the points 
.
It is not hard to show that
is proportional to
, where g is the genus of X.
As the latter expression is 
and this estimate cannot be
improved in general, we get that Kim's bound is best possible.
This argument also shows that Kim's bound implies the
Riemann hypothesis for curves over finite fields.
Another question is to bound the number of rational points. The following result was proved in [BV]:
The work of Caporaso et al. [CHM] shows that, in the number field case, uniform bounds for the number of rational points would follow from conjectures of Lang (see below). As some of these uniform bounds are false in conjectures of Lang on varieties of general type (see below). In [AV2] similar consequences of these conjectures are obtained in the function field case. As it turns out, counterexamples can be given to some of these consequences in characteristic p ([AV2]). Namely, we show that there are no uniform bounds for the number of points on curves that change genus or to the number of separable maps between curves depending on the gonality of the curve.
We conclude this section by mentioning the work of Denis [D2], where he finds all rational points in a certain class of curves which, from the point of view of Drinfeld modules, are the characteristic p analogues of the Fermat curves. These curves change genus in the above sense.
