  The question of the asymptotic form of the perturbation expansion in scalar
field theories is reconsidered. Renewed interest in the computation of terms in
the epsilon-expansion, used to calculate critical exponents, has been
frustrated by the differing and incompatible results for the high-order
behaviour of the perturbation expansion reported in the literature. We identify
the sources of the errors made in earlier papers, correct them, and obtain a
consistent set of results. We focus on phi^4 theory, since this has been the
most studied and is the most widely used, but we also briefly discuss analogous
results for phi^N theory, with N>4. This reexamination of the structure of
perturbation expansions raises issues concerning the renormalisation of
non-perturbative effects and the nature of the Feynman diagrams at large order,
which we discuss.
