  We show that if a sequence of dense graphs has the property that for every
fixed graph F, the density of copies of F in these graphs tends to a limit,
then there is a natural ``limit object'', namely a symmetric measurable
2-variable function on [0,1]. This limit object determines all the limits of
subgraph densities. We also show that the graph parameters obtained as limits
of subgraph densities can be characterized by ``reflection positivity'',
semidefiniteness of an associated matrix. Conversely, every such function
arises as a limit object. Along the lines we introduce a rather general model
of random graphs, which seems to be interesting on its own right.
