Problems in which immersed elastic structures interact with the surrounding fluid abound in biology, physics and engineering. Despite their scientific importance, analysis and numerical analysis of such problems are scarce or non-existent. In this talk, we consider the problem of an elastic filament immersed in a 2D or 3D Stokes fluid. We first discuss our recent results on the analysis of the immersed elastic interface problem in a 2D Stokes fluid (the Peskin problem). We prove well-posedness and immediate regularization of the elastic filament configuration, stability of steady states, criteria for global existence and discuss the implication of these results for numerical analysis. We will then discuss the immersed filament problem in a 3D Stokes fluid (the slender body problem). Here, it has not even been clear what the appropriate mathematical formulation of the problem should be. We propose a mathematical formulation for the slender body problem (the slender body PDE) and discuss well-posedness for the stationary version of this problem. Furthermore, we prove that the slender body approximation, introduced by Keller and Rubinow in the 1980's and used widely in the fluid-structure interaction community, provides an approximation to the slender body PDE with some error bound.