  In this paper we will review the main results concerning the issue of
stability for the determination unknown boundary portion of a thermic
conducting body from Cauchy data for parabolic equations. We give detailed and
selfcontained proofs. We prove that such problems are severely ill-posed in the
sense that under a priori regularity assumptions on the unknown boundaries, up
to any finite order of differentiability, the continuous dependence of unknown
boundary from the measured data is, at best, of logarithmic type.
