  We study the viscosity corrections to the growth rate of nucleating bubbles
in a first order phase transition in scalar field theory. We obtain the
non-equilibrium equation of motion of the coordinate that describes small
departures from the critical bubble and extract the growth rate consistently in
weak coupling and in the thin wall limit. Viscosity effects arise from the
interaction of this coordinate with the stable quantum and thermal fluctuations
around a critical bubble. In the case of 1+1 dimensions we provide an estimate
for the growth rate that depends on the details of the free energy functional.
In 3+1 dimensions we recognize robust features that are a direct consequence of
the thin wall approximation and give the leading viscosity corrections.These
are long-wavelength hydrodynamic fluctuations that describe surface waves,
quasi-Goldstone modes which are related to ripples on interfaces in phase
ordered Ising-like systems. We discuss the applicability of our results to
describe the growth rate of hadron bubbles in a quark-hadron first order
transition.
