The development of numerical solvers for fluid-elastic solid interaction problems has become particularly active since the 1980's. Among the most popular techniques are the Immersed Boundary method and the Arbitrary Lagrangian Eulerian method. We further mention the Fictitious Domain method, the Lattice Boltzmann method, the Level Set method, and the Coupled Momentum method. Until recently, only monolithic algorithms seemed applicable to blood flow simulations. These algorithms are based on solving the entire nonlinear, coupled problem as one monolithic system. They are, however, generally quite expensive in terms of computational time, programming time and memory requirements, since they require solving a sequence of strongly coupled problems using, e.g., the fixed point and Newton's methods. The multi-physics nature of the blood flow problem strongly suggests employing partitioned (or staggered) numerical algorithms, where the coupled fluid-structure interaction problem is separated into a fluid and a structure sub-problem. The fluid and structure sub-problems are integrated in time in an alternating way, and the coupling conditions are enforced asynchronously. These classical (Dirichlet-Neumann loosely-coupled) partitioned schemes work well for problems in aeroelasticity. Unfortunately, when the fluid and structure have comparable densities, which is the case in the blood flow application, the simple strategy of separating the fluid from the structure suffers from severe stability issues. To get around these difficulties, and to retain the main advantages of loosely-coupled partitioned schemes such as modularity, simple implementation, and low computational costs, several new loosely-coupled algorithms have been proposed recently. In this lecture we will present a short overview of the basic numerical strategies for solving fluid-elastic solid interaction in blood flow, and will present the core ideas behind a novel, stable, loosely-coupled scheme, recently introduced by the speaker and collaborators to numerically simulate a class of multi-physics problems arising in blood flow applications. A recent extension of the scheme to study fluid-multi-layered structure interaction problems in blood flow will also be presented.