The top quark forward-backward asymmetry ${A}_{\mathrm{FB}}^{t}$ measured at the Tevatron is above the standard model prediction by more than $2\ensuremath{\sigma}$ deviation, which might be a harbinger for new physics. In this work we examine the contribution to ${A}_{\mathrm{FB}}^{t}$ in two different new physics models: one is the minimal supersymmetric model without $R$ parity which contributes to ${A}_{\mathrm{FB}}^{t}$ via sparticle-mediated $t$ channel process $d\overline{d}\ensuremath{\rightarrow}t\overline{t}$; the other is the third-generation enhanced left-right model which contributes to ${A}_{\mathrm{FB}}^{t}$ via ${Z}^{\ensuremath{'}}$-mediated $t$ channel or $s$ channel processes. We find that in the parameter space allowed by the $t\overline{t}$ production rate and the $t\overline{t}$ invariant mass distribution at the Tevatron, the left-right model can enhance ${A}_{\mathrm{FB}}^{t}$ to within the $2\ensuremath{\sigma}$ region of the Tevatron data for the major part of the parameter space, and in optimal case ${A}_{\mathrm{FB}}^{t}$ can reach 12% which is slightly below the $1\ensuremath{\sigma}$ lower bound. For the minimal supersymmetric model without $R$ parity, only in a narrow part of the parameter space can the ${\ensuremath{\lambda}}^{\ensuremath{'}\ensuremath{'}}$ couplings enhance ${A}_{\mathrm{FB}}^{t}$ to within the $2\ensuremath{\sigma}$ region while the ${\ensuremath{\lambda}}^{\ensuremath{'}}$ couplings just produce negative contributions to worsen the fit.