We study various properties of a random graph R n , drawn uniformly at random from the class A n of all simple graphs on n labelled vertices that satisfy some given property, such as being planar or having tree-width at most k. In particular, we show that if the class A is 'small' and 'addable', then the probability that R n is connected is bounded away from 0 and from 1. As well as connectivity we study the appearances of subgraphs, and thus also vertex degrees and the numbers of

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... We see further that if A is 'smooth' then we can make much more precise statements for example concerning connectivity.