In 1994 Don Sarason asked the following question in a Springer problem book: characterize analytic functions $f$ and $g$ on the unit disk such that the product $T_fT_{\overline g}$ of two Toeplitz operators is bounded on the Hardy space $H^2$ or the Bergman space $A^2$. I will report on progress of this problem as well as two other companion problems in the contexts of the Hardy space, the Bergman space, and the Fock space.