  We discuss the strong-coupling expansion in Euclidean field theory. In a
formal representation for the Schwinger functional, we treat the off-diagonal
terms of the Gaussian factor as a perturbation about the remaining terms of the
functional integral. We first study the strong-coupling expansion in the \phi^4
theory and also quantum electrodynamics. Assuming the ultra-local
approximation, we examine the analytic structure of the zero-dimensional
generating functions in the complex coupling constants plane. Second, we
discuss the ultra-local generating functional in two idealized field theory
models. To control the divergences of the strong-coupling perturbative
expansion two different steps are used. First, we introduce a lattice structure
to give meaning to the ultra-local generating functional. Using an analytic
regularization procedure we discuss briefly how it is possible to obtain a
renormalized Schwinger functional associated with these scalar models, going
beyond the ultra-local approximation. Using the strong-coupling perturbative
expansion we show how it is possible to compute the renormalized vacuum energy
of a self-interacting scalar field, going beyond the one-loop level.
