  The goal of this paper is to develop the theory of weighted Diophantine
approximation of $p$-adic variables. Firstly, we establish complete analogues
of Khintchine's theorem and the Jarn\'ik-Besicovitch theorem for `weighted'
simultaneous Diophantine approximation in the $p$-adic case. Secondly, we
obtain a lower bound for the Hausdorff dimension of weighted simultaneously
approximable points lying on $p$-adic curves. One of the key tools in our
proofs is the Mass Transference Principle, including its recent extension due
to Wang and Wu. Our result for curves has a natural constraint on the exponents
of approximation and, in the case of polynomial curves, complements a theorem
of Bugeaud et al, which deals with large exponents.
