  The diagrammatic coaction maps any given Feynman graph into pairs of graphs
and cut graphs such that, conjecturally, when these graphs are replaced by the
corresponding Feynman integrals one obtains a coaction on the respective
functions. The coaction on the functions is constructed by pairing a basis of
differential forms, corresponding to master integrals, with a basis of
integration contours, corresponding to independent cut integrals. At one loop,
a general diagrammatic coaction was established using dimensional
regularisation, which may be realised in terms of a global coaction on
hypergeometric functions, or equivalently, order by order in the $\epsilon$
expansion, via a local coaction on multiple polylogarithms. The present paper
takes the first steps in generalising the diagrammatic coaction beyond one
loop. We first establish general properties that govern the diagrammatic
coaction at any loop order. We then focus on examples of two-loop topologies
for which all integrals expand into polylogarithms. In each case we determine
bases of master integrals and cuts in terms of hypergeometric functions, and
then use the global coaction to establish the diagrammatic coaction of all
master integrals in the topology. The diagrammatic coaction encodes the
complete set of discontinuities of Feynman integrals, as well as the
differential equations they satisfy, providing a general tool to understand
their physical and mathematical properties.
