  Invertibility of Toeplitz operators on the Bergman space and the related
Douglas problem are long standing open problems. In this paper we study the
invertibility problem under the novel geometric condition on the image of the
symbols, which relaxes the standard positivity condition. We show that under
our geometric assumption, the Toeplitz operator $T_\varphi$ is invertible if
and only if the Berezin transform of $|\varphi|$ is invertible in $L^{\infty}$.
It is well known that the Douglas problem is still open for harmonic functions.
We study a class of rather general harmonic polynomials and characterize the
invertibility of the corresponding Toeplitz operators. We also give a number of
related results and examples.
