  We present complete characterizations of Toeplitz operators that are complex
symmetric. This follows as a by-product of characterizations of conjugations on
Hilbert spaces. Notably, we prove that every conjugation admits a canonical
factorization. As a consequence, we prove that a Toeplitz operator is complex
symmetric if and only if the Toeplitz operator is $S$-Toeplitz for some
unilateral shift $S$ and the transpose of the Toeplitz operator matrix is equal
to the matrix of the Toeplitz operator corresponding to the basis of the
unilateral shift $S$. Also, we characterize complex symmetric Toeplitz
operators on the Hardy space over the open unit polydisc. Our results answer
the well known open question about characterizations of complex symmetric
Toeplitz operators.

Functional Analysis;Complex Variables;Operator Algebras