  We present a consistent picture of tunnelling in field theory. Our results
apply both to high-temperature field theories in four dimensions and to
zero-temperature three-dimensional ones. Our approach is based on the notion of
a coarse-grained potential U_k that incorporates the effect of fluctuations
with characteristic momenta above a given scale k. U_k is non-convex and
becomes equal to the convex effective potential for k --> 0. We demonstrate
that a consistent calculation of the nucleation rate must be performed at
non-zero values of k, larger than the typical scale of the saddle-point
configuration that dominates tunnelling. The nucleation rate is exponentially
suppressed by the action S_k of this saddle point. The pre-exponential factor
A_k, which includes the fluctuation determinant around the saddle-point
configuration, is well-defined and finite. Both S_k and A_k are k-dependent,
but this dependence cancels in the expression for the nucleation rate. This
picture breaks down in the limit of very weakly first-order phase transitions,
for which the pre-exponential factor compensates the exponential suppression.
