  We introduce and compare two approaches to equivariant homotopy theory in a
topological or ordinary Quillen model category. For the topological model
category of spaces, we generalize Piacenza's result that the categories of
topological presheaves indexed by the orbit category of a fixed topological
group $G$ and the category of $G$-spaces can be endowed with Quillen equivalent
model category structures. We prove an analogous result for any cofibrantly
generated model category and discrete group $G$, under certain conditions on
the fixed point functors of the subgroups of $G$. These conditions hold in many
examples, though not in the category of chain complexes, where we nevertheless
establish and generalize to collections an equivariant Whitehead Theorem \`{a}
la Kropholler and Wall for the normalized chain complexes of simplicial
$G$-sets.
