  Little Higgs models offer a new way to address the hierarchy problem, and
give rise to a weakly-coupled Higgs sector. These theories predict the
existence of new states which are necessary to cancel the quadratic divergences
of the Standard Model. The simplest version of these models, the Littlest
Higgs, is based on an $SU(5)/SO(5)$ non-linear sigma model and predicts that
four new gauge bosons, a weak isosinglet quark, $t'$, with $Q=2/3$, as well as
an isotriplet scalar field exist at the TeV scale. We consider the
contributions of these new states to precision electroweak observables, and
examine their production at the Tevatron. We thoroughly explore the parameter
space of this model and find that small regions are allowed by the precision
data where the model parameters take on their natural values. These regions
are, however, excluded by the Tevatron data. Combined, the direct and indirect
effects of these new states constrain the `decay constant' $f\gsim 3.5$ TeV and
$m_{t'}\gsim 7 $ TeV. These bounds imply that significant fine-tuning be
present in order for this model to resolve the hierarchy problem.
