Threshold phenomena in random structures

The physical theory of phase transition explains sudden changes of phase in materials that undergo gradual changes of some parameter like temperature. There are analogs of phase transition in the theory of random graphs, initiated by ErdGs and Renyi. This paper gives a nontechnical but precise account, without proofs, of some of the beautiful discoveries of Erdijs and RCnyi about threshold phenomena in graphs, describes an application of their methods to interval graphs, and gives some examples of threshold phenomena under other definitions of randomness and in combinatorial structures other than graphs. The paper offers some speculations on possible applications of random combinatorial structures to telecommunications, neurobiology, and the origin of life.