  We show that, assuming Vojta's height conjecture, the height of a rational
point on an algebraically hyperbolic variety can be bounded "uniformly" in
families. This generalizes a result of Su-Ion Ih for curves of genus at least
two to higher-dimensional varieties. As an application, we show that, assuming
Vojta's height conjecture, the height of a rational point on a curve of general
type is uniformly bounded. Finally, we prove a similar result for smooth
hyperbolic surfaces with $c_1^2 > c_2$.
