The subjects of elasticity and low-Reynolds-number flows intersect whenever viscous laminar flows occur in the presence of soft, deformable boundaries whose shapes are influenced by the flow. Since the flow is changed when the location of the boundary changes, there is feedback between the elastic and viscous flow problems. We have selected a variety of problems involving slender elastic filaments for which analytical calculations are possible. The shape of the filament follows by combining the description of the classical elastica with slender-body theory from low-Reynolds-number hydrodynamics. We consider only the linearized version of the equation for the elastica, and thus we solve a linear differential equation that includes terms representative of the elastic and viscous contributions. Some of the problems admit similarity solutions, and in most cases dimensional analysis is used to identify the important dimensionless parameters. In this way, a reader can find several problems that may be useful as exercises for a fluid mechanics or applied mathematics class or as a motivation for elegant calculations and scaling considerations that can form a bridge to real applications.