In nature, biology and industry, there are numerous examples of situations that can be modelled as Fluid-Structure Interaction problems (FSI for short). We are interested, in particular, in those cases in which some of the physical dimensions involved in the description of the problem are significantly smaller than the rest, thus resulting in a Fluid-Thin Structure Interaction problem (FTSI). Furthermore, in many cases, the deformation of the thin structure is constrained, or influenced, by the presence of another object, leading to Fluid-Thin Structure Interaction Contact problems (FTSIC). In this research we will develop and analyze new models describing these kinds of scenarios. These models will be supported by rigorous mathematical and numerical analysis results. Moreover, we will design numerical algorithms to be implemented in a computer, enabling the simulation and practical application of these new models.