In this thesis, we treat shape optimization for parabolic equations on time-dependent domains. As a theoretical foundation we extend the solution theory of parabolic boundary integral equations in the canonical Sobolev spaces from the cylindrical to the time-dependent case. The results imply existence and uniqueness of solutions. This is followed by a review of shape optimization theory on time-dependent domains, where we complement a few proofs which seem to be missing in the literature. Building on these foundations we give general formulae for shape gradients of functionals. These theoretical results are complemented by two numerical examples. The first example is concerned with a time-dependent shape detection problem, reformulated as a time-dependent shape optimization problem. The second example is a time-dependent one-phase Stefan problem in multiple dimensions, also reformulated as a shape optimization problem.