The dolfin-adjoint manual describes how to use dolfin-adjoint and the main features of dolfin-adjoint, and includes introductory examples. For more use cases, see the examples. For technical details about the framework used by dolfin-adjoint, see the pyadjoint documentation.
The mathematical background
Are you not that familiar with adjoints? Adjoints show up in many applications, and in many computational techniques. You might be interested in a mathematical introduction to adjoints and their applications.
Examples of dolfin-adjoint usage
We have implemented and documented a number of examples using dolfin-adjoint.
Many more examples of dolfin-adjoint usage can be found in the separate dolfin-adjoint applications repository on Bitbucket.