We consider the flavor-diagonal neutrino--two-photon vertex in the standard model. Whereas for massless neutrinos this vertex is of order {ital G}{sub {ital F}}{alpha}{ital p}{sub {nu}}/{ital m}{sub {ital W}}{sup 2}, for massive neutrinos a contribution to this amplitude of order {ital G}{sub {ital F}}{alpha}{ital m}{sub {nu}}/{ital m}{sub {ital e}}{sup 2} is identified and calculated. This contribution can be thought of as arising from a triangle graph containing electrons. The neutrino--two-photon vertex also gets a significant contribution from a second-order effect if the electromagnetic moments are anomalously large, as recently proposed to solve the solar-neutrino problem. The effects of the neutrino--two-photon vertex on neutrino-photon scattering and on photon pair annihilation into neutrinos are calculated for situations that may be of astrophysical or cosmological interest. The long-range force between a neutrino and a charge arising from two-photon exchange is also discussed.