The Titan-Saturn plasma interaction is studied using a global numerical kinetic model. The developed self-consistent three-dimensional quasi-neutral hybrid model is used to study Titan's interaction with supersonic and subsonic plasma flow. The subsonic flow case represents Titan in the Kronian magnetosphere during the Voyager-1 flyby. The supersonic and weakly supermagnetosonic flow cases are used to study a disturbed magnetospheric plasma flow event. In both cases the magnetic field is found to pile up against Titan and forming a magnetotail on the wake side. The subsonic flow case does not produce a bow shock in front of Titan, in contrast to the supersonic case. The magnetic field is found to have a Saturn facing side-anti Saturn facing side asymmetry in both cases. The subsonic case resulted in a twisting of the magnetotail toward Saturn that is qualitatively in agreement with the magnetic field observations made by Voyager 1.