  This article discusses random hypergraphs with varying hyperedge sizes,
admitting large hyperedges with size tending to infinity, and heavy-tailed
limiting hyperedge size distributions. The main result describes a threshold
for the random hypergraph to be connected with high probability, and shows that
the average hyperedge size suffices to characterise connectivity under mild
regularity assumptions. Especially, the connectivity threshold is in most cases
insensitive to the shape and higher moments of the hyperedge size distribution.
Similar results are also provided for related random intersection graph models.
