On the Hardy space, by means of an elegant and ingenious argument, Widom showed that the spectrum of a bounded Toeplitz operator is always connected  and Douglas showed that the essential spectrum of a bounded Toeplitz operator is also connected. On the Bergman space, McDonald and Sundberg showed that the essential spectrum of $T_/varphi$ is connected for $/varphi$ a harmonic function on $D$ if $/varphi$ is either real-valued or piecewise continuous on the boundary of the unit disk. They asked the problem whether the essential spectrum of a Toeplitz operator on the Bergman space with bounded harmonic symbol is connected. In my talk, I will present my joint work with Carl Sundberg to show examples that the spectrum and the essential spectrum of a Toeplitz operator with bounded harmonic symbol is disconnected.