This talk presents boundary integral equation (BIE)methods and related software for problems with moving boundaries. We will discuss two problems. In the first, we simulate the dynamics of highlyconcentrated vesicle suspensions in a Stokesian fluid. Such fluids are representative of real biological systems and are useful for studying the rheology of blood and other complex biofluids. In the second, we will discuss the problem of computing the equilibrium position of magnetic flux surfaces in magnetically confined plasmas. Such studies enable better designs for devices used for plasma confinement.We will discuss the challenges in developing BIE solvers for these problems\; such as accurate boundary discretization, efficient quadratures for singular and near-singular integration, need for reparameterization/remeshing and scalable implementation. We will present algorithms and software tools to address these issues.