  We propose a gauge invariant formulation of the effective potential in terms
of a gauge invariant order parameter, for the Abelian Higgs model. The one-loop
contribution at zero and finite temperature is computed explicitly, and the
leading terms in the high temperature expansion are obtained. The result is
contrasted to the effective potential obtained in several covariant
gauge-fixing schemes, and the gauge invariant quantities that can be reliably
extracted from these are identified. It is pointed out that the gauge invariant
effective potential in the one-loop approximation is complex for {\em all
values} of the order parameter between the maximum and the minimum of the tree
level potential, both at zero and non-zero temperature. The imaginary part is
related to long-wavelength instabilities towards phase separation. We study the
real-time dynamics of initial states in the spinodal region, and relate the
imaginary part of the effective potential to the growth rate of equal-time
gauge invariant correlation functions in these states. We conjecture that the
spinodal instabilities may play a role in non-equilibrium processes {\em
inside} the nucleating bubbles if the transition is first order.
