The standard conjectures (http://en.wikipedia.org/wiki/Standard_conjectures_on_algebraic_cycles) were pretty much designed to be used in this way (and then proved); but proofs are lacking, and some of the results now have non-conditional proofs. There are many related results in the theory of motives.

In number theory, Vandiver's conjecture (http://en.wikipedia.org/wiki/Vandiver%27s_conjecture) has begun to stand out, because of its connection with K-theory (which is another area in which there are large scale conjectures used in this way).