In this chapter, the mathematically simplest random graph, the Bernoulli random graph, is introduced. Each of the possible edges is present, independently, with the same probability, so that the model is one of a network entirely without structure. To start with, the structure of the graph in the neighbourhood of a point is investigated and is shown to be very similar to that of a branching process with Poisson-distributed offspring numbers. Explicit bounds on the accuracy of the approximation are derived, using the Poisson approximation techniques derived in Chapter 7. The classical threshold theorem for the existence of a giant component is then established; the precision of the neighbourhood approximation simplifies the proof. The counts of small subgraphs are then investigated, and a subgraph threshold theorem is proved. Finally, the distribution of the length (in graph distance) of the shortest path between two vertices is investigated. These grow logarithmically with the number of points, if the expected degree of a vertex is kept constant. Once again, the approximation of the neighbourhood structure is a key element in the proofs, and the statement of the main theorem involves the Laplace transform of the distribution of the limit random variable associated with the approximate branching process.