  We evaluate the observational constraints on the spectral index $n$, in the
context of the $\Lambda$CDM hypothesis which represents the simplest viable
cosmology. We first take $n$ to be practically scale-independent. Ignoring
reionization, we find at a nominal 2-$\sigma$ level $n\simeq 1.0 \pm 0.1$. If
we make the more realisitic assumption that reionization occurs when a fraction
$f\sim 10^{-5}$ to 1 of the matter has collapsed, the 2-$\sigma$ lower bound is
unchanged while the 1-$\sigma$ bound rises slightly. These constraints are
compared with the prediction of various inflation models. Then we investigate
the two-parameter scale-dependent spectral index, predicted by running-mass
inflation models, and find that present data allow significant scale-dependence
of $n$, which occurs in a physically reasonable regime of parameter space.
