  Starting from generic quantum effects at the Planck scale M_P, we find that
the renormalization group running of the cosmological constant (CC) at low
energies is possible if there is a smooth decoupling of all massive particles
from M_P to the mass of the lightest neutrino. We discuss the theoretical
implications of this running for the ``old'' and ``new'' cosmological constant
problems. Interestingly enough, the CC running implies a strong relationship
between quantum field theory and cosmology, which should be observable in the
near future in experiments such as SNAP through the measurement of a cubic
redshift dependence of the CC.
