  We study the the following question in Random Graphs. We are given two
disjoint sets $L,R$ with $|L|=n=\alpha m$ and $|R|=m$. We construct a random
graph $G$ by allowing each $x\in L$ to choose $d$ random neighbours in $R$. The
question discussed is as to the size $\mu(G)$ of the largest matching in $G$.
When considered in the context of Cuckoo Hashing, one key question is as to
when is $\mu(G)=n$ whp? We answer this question exactly when $d$ is at least
four. We also establish a precise threshold for when Phase 1 of the Karp-Sipser
Greedy matching algorithm suffices to compute a maximum matching whp.
