Let H be a fixed graph on v vertices. For an n-vertex graph G with n divisible by v, an H-factor of G is a collection of n/v copies of H whose vertex sets partition V (G).

In this work, we consider the threshold thH(n) of the property that an Erds-Rényi random graph (on n points) contains an H-factor. Our results determine thH(n) for all strictly balanced H.

The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random k-uniform hypergraph, solving the well-known Shamir's problem.