In this work, the authors prove two theorems on the boundedness of bilinear pseudodifferential operators of the form
under suitable growth conditions on the symbol and its derivatives. Moreover, they explain how their methods are linked to the boundedness theorems of A. P. Calderón and R. Vaillancourt [J. Math. Soc. Japan 23, 374–378 (1971; Zbl 0203.45903)] and I. L. Hwang [Trans. Am. Math. Soc. 302, 55–76 (1987; Zbl 0651.35089)] for linear pseudodifferential operators.