  We have studied the effect of a non-vanishing neutrino magnetic moment \
($\mu_{\nu}$) on the $\nu_{\rm x}$ ({x=$e,\mu,\tau$}) elastic scattering off
electrons for the Super-Kamiokande detector. The bounds on the $\mu_{\nu}$ we
have obtained are comparable to that extracted from laboratory experiments.
Furthemore, we outline the potential of the Borexino experiment which may be
sensitive to neutrino magnetic moments $\lsim 10^{-10}\mu_B$. In our analysis
we have considered both cases of Majorana and Dirac neutrinos.
