Abstract:  In 1964 Brown and Halmos obtained characterizations for some fundamental algebraic properties of Toeplitz operators on the Hardy space over the disk. One of their results asserts that two Toeplitz operators can commute only in the trivial cases. Such a theorem of Brown-Halmos initiated study on zero/compact (semi-)commutators of Toeplitz operators on various settings. In this talk I will survey what have been done in that direction for the past 6 decades. During the presentation, the spaces will be restricted to (one-or-several-variable) classical spaces such as the Hardy space, the Bergman space and the Fock space. Some problems, which are still open in the case of the Fock space, will be announced. I believe that anyone who has graduate-level knowledge about the Fock space and Toeplitz operator may challenge the problems.