  For a nondegenerate $r$-graph $F$, large $n$, and $t$ in the regime $[0,
c_{F} n]$, where $c_F>0$ is a constant depending only on $F$, we present a
general approach for determining the maximum number of edges in an $n$-vertex
$r$-graph that does not contain $t+1$ vertex-disjoint copies of $F$. In fact,
our method results in a rainbow version of the above result and includes a
characterization of the extremal constructions.
  Our approach applies to many well-studied hypergraphs (including graphs) such
as the edge-critical graphs, the Fano plane, the generalized triangles,
hypergraph expansions, the expanded triangles, and hypergraph books. Our
results extend old results of Simonovits~\cite{SI68} and Moon~\cite{Moon68} on
complete graphs and can be viewed as a step towards a general density version
of the classical Corr\'{a}di--Hajnal Theorem~\cite{CH63}.
