Recent experimental and theoretical data on the rare electromagnetic conversion decays of light mesons ( A$\to$B$\gamma_v\to$B$l^+l^-$) are reviewed. In these decays, the photon is a virtual photon $(\gamma_v)$ and converts into a lepton pair $l^+l^-$. Study of the effective-mass spectrum of the lepton pairs reveals the dynamic transition form factor $F_{\mathbf{AB}}(q^2=m^2_{l^+l^-})$, which characterizes the electromagnetic structure of the A$\to$B$\gamma$ transition vertex. Such studies have been undertaken for the decays $\pi^0\to e^+e^-\gamma$, $\eta\to\mu^+\mu^-\gamma$, $\eta'\to\mu^+\mu^-\gamma$, $\omega\to\pi^0\mu^+\mu^-$, and several similar processes. Analysis of the results reveals that the general behavior of the transition form factors of the neutral mesons agrees with vector dominance and with various versions of the quark model. There are, on the other hand, some quantitative deviations from vector dominance in the decays $\pi^0\to e^+e^-\gamma$ and $\omega\to\pi^0\mu^+\mu^-$. Data on the electromagnetic leptonic processes of higher orders, such as the decays $\eta\to\mu^+\mu^-$ and $\pi^0\to e^+e^-$, and their relationship with the structure of mesons are discussed. Various experimental searches are also discussed. The future outlook for such research is assessed.