When I was a graduate student in math (mid-late eighties and early nineties) the arena was dominated by a few grand projects: for instance, Misha Gromov's hyperbolic groups, which spread into many seemingly heterogeneous domains (such as combinatorial group theory, to name just one), and Bill Thurston's classification of low-dim manifolds.
A few years have passed, and I must admit that, aside my pet domains, such as categorical algebra and applied logic, I have not kept up with innovations.
I am pretty sure, though, that new trends, generated by some core new ideas, are leading contemporary math research, or substantial portions thereof. One such idea I know already, namely Voevodsky's homotopical foundations of logics, which brings together abstract homotopy theory and type theory.
What else?
PS I am not interested (at least not directly) in new solutions to old problems, unless these solutions are the germs of (and conducive to) new directions.
Rather, I would hear about new research projects which unify previously disparate fields, or create entirely new ones, or shed lights into old domains in a radically new way (in other words, I am interested in what Thomas Kuhn called paradigm shifts). The ors are of course not mutually exclusive.
Addendum: It looks like there is an ongoing debate on whether this question should be kept open or not. As I have already answered below, I submit entirely and with no reservations to the policy of this respectable forum, whatever the outcome. As a dweller of MO, though, I am entitled to my personal opinion, and this is to keep my question in the air. Nevertheless, I am well aware of the potential risks of either turning it into -what I like best of math right now- or generic answers that provide no meat for this community. Therefore, allow me to add a clarification:
My wish is the following- to get to know from informed individuals which new paradigm-shifting trends are currently under way.
To this effect I appreciate all the answers so far, but I invite everybody to provide some level of details as to why they have mentioned a specific idea and/or research project (some folks have already done it). For instance, one answer was tropical math, which indeed seems to have risen to prominence in recent years. It would be nice to know which areas it does impact and in which way, which core ideas it is founded on, which threads it brings together, etc. The same applies of course to all other proposals.
Best Answer
The Langlands program. It goes back to the sixties, but in the last years, with the proof of the fundamental lemma by Ngô Bảo Châu and with several results in the local case, it became one of the most active area in number theory, and I think there is no hope to finish the job in the next, say, 50 years.