  We extend and improve several known results about the boundedness of the
bilinear pseudo-differential operators. First, we consider symbol classes of
$S_{0,0}$-type that generalize the bilinear H\"ormander class
$BS^{m}_{0,0}(\mathbb{R}^n)$ and characterize those classes for which the
corresponding bilinear operators are bounded in Wiener amalgam spaces.
Secondly, using the results for Wiener amalgam spaces, we prove sharp results
that certain $L^q$ or $L^{q,\infty}$ integrability of the symbol and its
derivatives with respect to the frequency variables implies boundedness of the
corresponding bilinear operator in Lebesgue spaces. As an application, we give
a generalization of the result about the $L^2 \times L^2 \to L^{1}$ boundedness
of bilinear operators due to Grafakos, He, and Slav\'ikov\'a (2020) to $L^{p_1}
\times L^{p_2} \to L^{p}$.
