There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily
so well known) include
The Clay Mathematics Institute (http://www.claymath.org/millennium/) of Cambridge, Massachusetts (CMI) has named seven "Millennium Prize Problems,"
selected by focusing on important classic questions in mathematics that have resisted
solution over the years. A $7 million prize fund has been established for the solution
to these problems, with $1 million allocated to each. The problems consist of the
Riemann hypothesis, Poincaré
conjecture, Hodge conjecture, Swinnerton-Dyer
Conjecture, solution of the Navier-Stokes equations, formulation of Yang-Mills
theory, and determination of whether NP-problems are
actually P-problems.
In 1900, David Hilbert proposed a list of 23 outstanding problems in mathematics (Hilbert's problems), a number of which have
now been solved, but some of which remain open. In 1912, Landau proposed four simply
stated problems, now known as Landau's problems,
which continue to defy attack even today. One hundred years after Hilbert, Smale
(2000) proposed a list of 18 outstanding problems.
K. S. Brown, D. Eppstein, S. Finch, and C. Kimberling maintain webpages of unsolved problems in mathematics. Classic texts on unsolved problems
in various areas of mathematics are Croft et al. (1991), in geometry,
and Guy (2004), in number theory.