Open Problems....

Finding interesting problems to work on is always challenging for a professional researcher. Usually a nice problem to work on should be

(1) interesting to you !!
(2) at least of some interest to the community

People may argue about the relevance of (2). The way I see it is that research is often like being at a very large party. There are many conversations going on all the time, and
you can either start one, or join in an ongoing discussion. You could also stand in a corner and talk to yourself, but it's not always much fun, and you may not always get people to listen.

So how does one find out what problems are of interest to the community ? Some factors:
(a) is the problem of relevance to many other problems (a "hub" problem)
(b) is it something that has been around for a while and is unsolved (an "authority" problem, to stretch a PageRank metaphor).

For some time now, the computational geometry has maintained a list of open problems that by virtue of their existence satisfy both of the above conditions.
The Open Problems Project is edited by Erik Demaine, Joe Mitchell, and Joe O'Rourke, and has grown from its original set of 30 to 56 problems now.

Some of these problems are quite hard and have been around for a while; others may just need the right idea to be cracked. Knowing which is which is the $1,000,000 question :)