  In this paper, we studied the $\rho \gamma^* \to \pi$ and $\rho\gamma^*\to
\rho$ transition processes and made the calculations for the $\rho\pi$
transition form factor $Q^4 F_{\rho\pi}(Q^2)$ and the $\rho$ meson
electromagnetic form factors, $F_{\rm LL, LT,TT}(Q^2)$ and $F_{1,2,3}(Q^2)$, by
employing the perturbative QCD (PQCD) factorization approach. For the $\rho
\gamma^* \to \pi$ transition, we found that the contribution to form factor
$Q^4 F_{\rho\pi}(Q)$ from the term proportional to the distribution amplitude
combination $\phi^T_{\rho}(x_1)\phi^P_{\pi}(x_2)$ is absolutely dominant, and
the PQCD predictions for both the size and the $Q^2$-dependence of this form
factor $Q^4 F_{\rho\pi}(Q^2)$ agree well with those from the extended ADS/QCD
models or the light-cone QCD sum rule. For the $\rho \gamma^* \to \rho$
transition and in the region of $Q^2\geq 3$ GeV$^2$, further more, we found
that the PQCD predictions for the magnitude and their $Q^2$-dependence of the
$F_1(Q^2)$ and $F_2(Q^2)$ form factors agree well with those from the QCD sum
rule, while the PQCD prediction for $F_3(Q^2)$ is much larger than the one from
the QCD sum rule.
