Fluid-structure interaction (FSI) problems have received increasing attention from various fields of applied sciences and engineering, with applications ranging from geophysics to bio-medicine. The design of computational techniques for FSI depends on the choice of the discrete representation of the domains, with two main categories: ``boundary-fitted'' or ``non-boundary-fitted'' methods. In the boundary-fitted case, the fluid-structure interface is represented explicitly, and the fluid mesh is deformed together with the structure mesh, typically in an arbitrary Eulerian-Lagrangian framework. Such methods are accurate, but large deformation in the structure may lead to heavy distortions in the fluid mesh and consequently to numerical issues and the necessity of remeshing. Methods that fall into the non-boundary-fitted category have the fluid and structure meshes represented separately; hence they are non-matching and consequently require higher resolution for a comparable accuracy. We present an immersed technique for the numerical solution FSI problems. Here, the fluid and structure are coupled in the overlapping volume, while different structures in contact are coupled on the surface using mortar-based techniques. In particular, we employ dual Lagrange multipliers, which, within the nonlinear solution procedure, allow us to transfer discrete fields with standard matrix-vector multiplication. This solution strategy is specifically designed to solve the contact problem with non-smooth methods; here, fluid and structure sub-problems are solved in a segregated and iterative way. We illustrate our general algorithmic framework and main parallel computing tools and discuss recent developments. We present an application of our methodology to a biomedical scenario. We simulate the full dynamics of a bio-prosthetic heart valve. We model the blood-valve interaction, the blood-aortic wall interaction, and the contact among leaflets during the valve closure. Additionally, we illustrate examples of our coupling techniques applied to different application fields.