  I present a sequence of non-perturbative approximate solutions for scalar
$\phi^4$ theory for arbitrary interaction strength, which contains, but allows
to systematically improve on, the familiar mean-field approximation. This
sequence of approximate solutions is apparently well-behaved and numerically
simple to calculate since it only requires the evaluation of (nested) one-loop
integrals. To test this resummation scheme, the case of $\phi^4$ theory in 1+1
dimensions is considered, finding approximate agreement with known results for
the vacuum energy and mass gap up to the critical point. Because it can be
generalized to other dimensions, fermionic fields and finite temperature, the
resummation scheme could potentially become a useful tool for calculating
non-perturbative properties approximately in certain quantum field theories.
