  Following Georgi's unparticle scheme, we examine the effective couplings
between neutrinos and unparticle operators. As an immediate consequence,
neutrinos become unstable and can decay into the unparticle stuff. Assuming the
dimension transmutation scale is around $\Lambda^{}_{\cal U} \sim 1 ~{\rm
TeV}$, we implement the cosmological limit on the neutrino lifetime to
constrain the neutrino-unparticle couplings for different scaling dimensions
$d$. In addition, provided that the electron-unparticle coupling is restricted
due to the precise measurement of the anomalous magnetic moment of electron, we
calculate the unparticle contribution to the neutrino-electron elastic
scattering. It is more important to jointly deal with the couplings of the
unparticle to the standard model particles rather than separately. Taking into
account both electron- and neutrino-unparticle couplings, we find that the
scaling dimension of the scalar unparticle should lie in the narrow range $1 <
d < 2$ by requiring the observables to be physically meaningful. However, there
is no consistent range of $d$ for the vector unparticle operator.
