  We investigate families of graphs and graphons (graph limits) that are
defined by a finite number of prescribed subgraph densities. Our main focus is
the case when the family contains only one element, i.e., a unique structure is
forced by finitely many subgraph densities. Generalizing results of Turan,
Erdos-Simonovits and Chung-Graham-Wilson, we construct numerous finitely
forcible graphons. Most of these fall into two categories: one type has an
algebraic structure and the other type has an iterated (fractal-like)
structure. We also give some necessary conditions for forcibility, which imply
that finitely forcible graphons are "rare", and exhibit simple and explicit
non-forcible graphons.
