Abstract: A complex symmetric  operator refers to a linear operator with a symmetric matrix representation  on a  Hilbert space. We review the main research  advances  and  several open problems of complex symmetric operators in recent years, involving special complex symmetric operators, reducing subspaces, the norm closure problem, and algebraic properties  and so on.

Key words: complex symmetric operator, skew symmetric operator, Toeplitz operator, truncated Toeplitz operator, weighted shift, partial isometry, reducing subspace