Abstarct: The interaction of fluid/deformable structures is studied in the preset research because of its importance in natural and industrial fields. Suitable potency of the immersed boundary-lattice Boltzmann method (IB_LBM) persuade the researchers to use the method in different physics. In IB_LBM; the flow filed is covered by non-body-fitted Cartesian mesh. For solving the fluid flow field, the lattice Boltzmann method is employed which contains a simple set of equations. In recent years, the Lattice Spring method (LSM) is also used for analysis of deformable bodies instead of common continuum equations. Moreover, LSM shows good efficiency simulation of elastic bodies. This method has also many important applications in modeling of the biological systems like red blood cells. In many technical applications; it is necessary to simulate the viscoelastic solids. Many of biological particles and bodies have the behavior of viscoelastic solids. In the following chapters, history of the recent studies, a descxription of the immersed boundary method, the Lattice Boltzmann method and IB_LBM_LSM are provided and the constitutive equations are analyzed. In addition, the Lattice Spring method is also presented and a new network of ‘’Mass-Spring-Dashpot’’ is proposed. The important innovation of the present study is the simulation of viscoelastic solids with a network of ‘’Mass-Spring-Dashpot’’, with ability of changing the present model to simulate the simple elastic materials, Kelvin viscoelastic networks, Maxwell viscoelastic networks and also the rigid networks. By using the Lattice spring method for simulation of the elastic filaments, the Young's module, bending and shear modules and Possion ratio can be tuned. The network of Mass-Spring-Dashpot can be employed to obtain the important rheometric properties of the viscoelastic solids. The main purpose of this research is simulation of elastic and viscoelastic filaments in the presence of fluid flow. The operation of Mass-Spring-Dashpot network in this problem is discussed by considering oscillation patterns, frequency and amplitude of flapping, drag and lift coefficients of filament and geometrical parameters. One of the applications of deformable filaments in fluid mechanics is drag reduction. In this way, an elastic thin filament is located at the back of a rigid cylinder, and the effects of distance between filament and the cylinder on drag reduction are studied. Mass (or density) of filament is a very important parameter for detection of regime of oscillation and the amplitude of oscillation of the filament. Moreover, the length of filament have important effects on the regime of flapping. Increasing the length and mass of the filament leads to a change in the filament oscillation pattern. This pattern is a chaotic regime with significant increase in the amplitude of oscillation. Viscosity of viscoelastic filament is another important parameter for change of regime. Increase of the viscoelastic filament viscosity can regulate the oscillation pattern.