  For the theory of a single scalar field $\varphi$ with a quartic potential
$V(\varphi)$, we find semi-analytic expressions for the Euclidean action in
both four and three dimensions. The action in four dimensions determines the
quantum tunneling rate at zero temperature from a false vacuum state to the
true vacuum state; similarly, the action in three dimensions determines the
thermal tunneling rate for a finite temperature theory. We show that for all
quartic potentials, the action can be obtained from a one parameter family of
instanton solutions corresponding to a one parameter family of differential
equations. We find the solutions numerically and use polynomial fitting
formulae to obtain expressions for the Euclidean action. These results allow
one to calculate tunneling rates for the entire possible range of quartic
potentials, from the thin-wall (nearly degenerate) limit to the opposite limit
of vanishing barrier height. We also present a similar calculation for
potentials containing $\varphi^4 \ln \varphi^2$ terms, which arise in the
one-loop approximation to the effective potential in electroweak theory.
