Which set is the numerical range of an operator?

In this survey paper, we consider the problem of which nonemptybounded convex subset of the complex plane is the numerical range of somebounded linear operator on a complex separable Hilbert space. We startin Section 1 with general operators and move subsequently to operatorsin certain special classes such as normal and hyponormal operators inSection 2, Toeplitz operators in Section 3, Hankel operators in Section 4,compact operators in Section 5, nilpotent operators and roots of identityin Section 6, S_n-matrices and companion matrices in Section 7, and, finally,nonnegative matrices in Section 8. The known main results will be brieflysketched, which are interspersed with relevant unsolved problems.