  Graphical functions are single-valued complex functions which arise from
Feynman amplitudes. We study their properties and use their connection to
multiple polylogarithms to calculate Feynman periods. For the zig-zag and two
more families of phi^4 periods we give exact results modulo products. These
periods are proved to be expressible as integer linear combinations of
single-valued multiple polylogarithms evaluated at one. For the larger family
of 'constructible' graphs we give an algorithm that allows one to calculate
their periods by computer algebra. The theory of graphical functions is used to
prove the zig-zag conjecture.
