We report on the calculation of the cross sections for the production of positive-charge-conjugation states such as ${\ensuremath{\pi}}^{0}$, $\ensuremath{\eta}$, ${e}^{+}{e}^{\ensuremath{-}}$, ${\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$, and ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ by a two-photon mechanism in ${e}^{\ensuremath{-}}{e}^{+}$ and ${e}^{\ensuremath{-}}{e}^{\ensuremath{-}}$ collisions. We give the precise relationship of the process $e+e\ensuremath{\rightarrow}e+e+X (X \mathrm{is}\mathrm{any} C=+ \mathrm{state})$ to the corresponding two-photon annihilation process $\ensuremath{\gamma}+\ensuremath{\gamma}\ensuremath{\rightarrow}X$, as well as a careful derivation of the equivalent-photon approximation. In the case of the ${\ensuremath{\pi}}^{0}$ production, we have found that, for the beam energy $E$ in the 1-3 GeV range, the exact total cross section is 20-30% larger than the one calculated previously in the equivalent-photon approximation. However, the introduction of form factors cuts down the exact total cross section, reducing it to within 10% of the equivalent-photon-approximation result. For $\ensuremath{\eta}$ production the agreement is even better. Thus it appears that the use of the equivalent-photon approximation is justified in most cases. We discuss detailed angular distributions in this approximation for the case of ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ production. One important problem which cannot be adequately studied in the equivalent-photon approximation is the degree of noncoplanarity of the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ (and the ${\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$) pair. We have studied this problem using the exact formula and found that, for $E=1$ GeV, typically 40-50% of pion pairs will be produced with the noncoplanarity angle greater than 12\ifmmode^\circ\else\textdegree\fi{}. We discuss the general structure of the $\ensuremath{\gamma}+\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}+{\ensuremath{\pi}}^{\ensuremath{-}}$ amplitude as well as a simple model incorporating the $\ensuremath{\sigma}$ meson. We also give a rough estimate of multihadron production cross sections.