We present a novel approach to the generation of random hypergraphs in user-specified domains. Our method extends an algorithm of Mairson (1994) for generating strings in context-free languages to the setting of context-free hypergraph languages specified by hyperedge replacement grammars. Generating (or “sampling") graphs and hypergraphs according to a given probability distribution is a problem that finds application in testing algorithms and programs working on graphs. Other potential application areas include molecular biology and post-quantum cryptography. We show that if the input grammar is non-ambiguous, the generated samples are uniformly distributed. The only requirements for our method are that the properties sought for the generated hypergraphs are representable by a hyperedge replacement language and that, to guarantee a uniform distribution, a non-ambiguous grammar is used as input. We also show that our method generates a random hypergraph of size n in time O(n2).