  Butcher series, also called B-series, are a type of expansion, fundamental in
the analysis of numerical integration. Numerical methods that can be expanded
in B-series are defined in all dimensions, so they correspond to
\emph{sequences of maps}---one map for each dimension. A long-standing problem
has been to characterise those sequences of maps that arise from B-series. This
problem is solved here: we prove that a sequence of smooth maps between vector
fields on affine spaces has a B-series expansion if and only if it is
\emph{affine equivariant}, meaning it respects all affine maps between affine
spaces.
