I wonder what you all think are some questions that our current understanding of math can't really answer but which, in the reasonably near future (Let's call this "before you retire"), we will be able to answer? This is a deliberately broad question. You can speculate on particular conjectures, on developing interrelations between disciplines, on methods, applications or anything you want. Try to describe a little bit what sort of program you see leading up to these advances.
The only area I can really attest to is that of complex systems (large networks of interacting agents, in general)). Right now, this isn't really a mathematical discipline. Non-linear dynamics and applied probability sometimes touch on questions of complex systems, but right now nearly all study in the field is of a holistic and decidedly non-rigorous sort. NOTE: If you're an applied probabilist, don't think I'm forgetting about you - it's just that there's generally a huge gulf between our ability to simulate these systems very approximately and our ability to define their behavior rigorously.
I believe that, within the next 20 years, we'll see a modest revolution in this field. We'll be able to bring in tools from the fields of probability and statistical mechanics and create new tools such that complex systems will allow us to understand certain collective human networks (genetic networks, contagion, traffic, urban growth are probably some of the most achievable) on the same level that we understand systems of particles today (or even that we understood systems of particles 50 years ago). In its early stages, this probably still won't be rigorous, we'll rely on hybrid techniques from machine learning, stochastic processes and domain-specific techniques to deal with each sort of problem. By the end of a couple more decades, however, we'll have made leaps forward in our ability to rigorously describe, understand and predict complex systems.