  Consider a smooth, geometrically irreducible, projective curve of genus $g
\ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely
many rational points by the Mordell Conjecture, a theorem of Faltings. We show
that the number of rational points is bounded only in terms of $g$, $d$, and
the Mordell-Weil rank of the curve's Jacobian, thereby answering in the
affirmative a question of Mazur. In addition we obtain uniform bounded, in $g$
and $d$, for the number of geometric torsion points of the Jacobian which lie
in the image of an Abel-Jacobi map. Both estimates generalize our previous work
for $1$-parameter families. Our proof uses Vojta's approach to the Mordell
Conjecture, and the key new ingredient is the generalization of a height
inequality due to the second- and third-named authors.
