The term "slender" refers to structures with a very high ratio between their longitudinal lenght and their transverse dimensions, typically, a cylinder with an height significantly larger than its radius. Because of this particular geometry, many models have been developed to provide a simplified description of the kinematics and dynamics of the structure. A standard approach in this context is to account for the distribution of forces and deformations only along the centerline. Consequently, the velocity fields and equilibrium equations of the structure are described in a one-dimensional (1D) setting. However, when a slender structure is immersed in a three-dimensional (3D) fluid, enforcing kinematic and dynamic coupling conditions on a 1D domain requires the introduction of a double trace operator (codimension 2) which demands regularity for the solution within the fluid domain, a condition which is generally not satisfied a priori. In this talk, I will introduce and analyse a new mathematically sound approach for modelling and solving 3D-1D fluid-structure interaction problems. The main idea is to combine a fictitious domain approach with the projection of the kinematic constraint onto a finite-dimensional space defined along the structure's centerline. The discrete formulation is based on the finite element method and a semi-implicit treatment of the Dirichlet-Neumann coupling conditions, employing a partitioned procedure for the resolution of the fluid-structure interaction problem.