An open problem is a problem that can be formally stated and for which a solution is known to exist but which has not yet been solved. It is common in graduate schools to point out open problems to students. However, Fermat's last theorem and the Four color theorem are two notable open problems that have been closed or solved by faculty members, as they would have more expertise and resources than students.
Important open problems exist in many fields, such as in the field of theoretical computer science, computer scheduling, and real-time computing.
Advice to Research Students - Some tips for solving problems in number theory, collected by the Cardiff University group.
L-functions and Random Matrix Theory - Conjectures and open problems concerning L-functions, focussing on the areas in which there has been recent progress using results from Random Matrix Theory. Maintained at AIM.
Mathematician's Secret Room - Unsolved problems in Number Theory. English and Japanese text by Hisanori Mishima.
Number Theory Conjectures - A collection of easily stated conjectures which are still open. Each conjecture is stated along with a collection of references.