This thesis represents a numerical method to solve fluid-structure interaction (FSI) prob- lems where the fluid exhibits a non-Newtonian behavior. Oldroyd-B constitutive equa- tions is used to model Viscoelastic fluid and Generalized Newtonian Fluids are used to model Shear-thinning/Shear-thickening fluids. For FSI solution, a semi-implicit partitioned method is used which separates the fluid pressure term and strongly couples it to the struc- ture, while the remaining fluid terms are only weakly coupled. Numerical tests provides information regarding the physical behaviour of the fluid and structural response of the elastic domain in non-Newtonian fluid-structure interaction problems