Abstract: In many computational biology problems, one must adjust a designated protein fragment to connect its two termini to the rest of the protein (assumed fixed) in a way that is consistent with the kinematics of the fragment. For example, in structure determination from X-ray crystallographic data, existing software often builds an initial partial structure; missing fragments must then be filled in to bridge the gaps between resolved termini. Similarly, in homology modeling, only parts of the protein structure can be reliably inferred from homologous proteins, while remaining fragments of unknown structures have to be inserted. The problem also occurs during ab initio structure prediction when one attempts to tweak protein fragments into energetically more favorable conformations. The underlying problem -- finding three-dimensional conformations of an inner fragment of a protein chain that are kinematically consistent with the rest of the chain --– is often referred to as the loop closure problem. In this talk I will describe various algorithms that we are implementing to efficiently sample conformations of protein loops (or fragments). I will also talk about new theoretical results on the topological structure of the conformational space of a 3-amino-acid fragment.