Recently, the US National Academy of Engineering announced top 14 grand challenges we face in the 21st century. They have focused on the problems that should be (can be) solved (solvable) in the near future.
-Advance health informatics
-Engineer better medicines
-Make solar energy economical
-Provide access to clean water
-Reverse-engineer the brain
-Advance personalized learning
-Engineer the tools of scientific discovery
-Manage the nitrogen cycle (What this basically means is to find ways to reproduce nitrogen to maintain the bio-geochemical cycle which is being altered by increased use of fertilizers and industrial activities. Reduction in nitrogen in air leads to, among other things, smog, acid rain, polluted drinking water and global warming.
-Provide energy from fusion
-Secure cyberspace (identity thefts, viruses, etc.)
-Develop carbon sequestration methods (methods to capture carbon dioxide produced from vehicles and factories from burning fossil fuels, a major culprit of global warming)
-Enhance virtual reality
-Prevent nuclear terror ( I think this is more political than technological. We already have the technology in place; it's a matter of how we use it only for purposes that benefit humankind.)
-Restore and improve urban infrastructure
There's a good introduction about this in their website.
I personally think the following are especially important (from a technological point of view but not political):
-"Economical" clean energy sources to substitute for fossil fuels (fast diminishing)
-Methods to remove carbon dioxide and cycle nitrogen to control global warming
-Methods to prevent, detect, recover from, cyber attacks. With more and more people/businesses do their financial transactions online, we need effective methods to protect them from malicious attackers.