We study boundedness properties of pseudodifferentail operators of Weyl calculus on modulation spaces introduced by Feichtinger. These spaces play a similar role in Gabor analysis as Besov spaces for wavelets. Recently, Grochenig and Heil have shown that the operators with symbols in $M_{\infty, 1}$ are bounded on all modulation spaces, thus proving the importance of modulation spaces in the theory of pseudodifferential operators. We further study the boudedness properties of pseudodifferential operators by introducing classes of symbols related to modulation spaces. Our approach may serve as an approximation to the results of Grochenig and Heil. We also show how it is related to the work of other authors.