A k-uniform hypergraph H = (V, E) is called ℓ-orientable if there is an assignment of each edge e ∈ E to one of its vertices v ∈ e such that no vertex is assigned more than ℓ edges. Let Hn,m,k
be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the ℓ-orientability of Hn,m,k
for all k ⩾ 3 and ℓ ⩾ 2, that is, we determine a critical quantity c*k,ℓ
such that with probability 1 − o(1) the graph Hn,cn,k
has an ℓ-orientation if c < c*k,ℓ
, but fails to do so if c > c*k,ℓ
Our result has various applications, including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.