  The Toeplitz operator acting on the Bergman space $A^{2}(\mathbb{D})$, with
symbol $\varphi$ is given by $T_{\varphi}f=P(\varphi f)$, where $P$ is the
projection from $L^{2}(\mathbb{D})$ onto the Bergman space. We present some
history on the study of hyponormal Toeplitz operators acting on
$A^{2}(\mathbb{D})$, as well as give results for when $\varphi$ is a
non-harmonic polynomial. We include a first investigation of Putnam's
inequality for hyponormal operators with non-analytic symbols. Particular
attention is given to unusual hyponormality behavior that arises due to the
extension of the class of allowed symbols.
