Abstarct: Generally, motion and shape of the drop falling under gravity in an immiscible fluid has become a benchmark problem in fluid dynamics and has a wide range of application in petroleum, medicine processing, mextals extraction, power plant and heat exchanger. In this thesis, analytical and experimental for creeping motion of viscoelastic drop falling through a viscose Newtonian fluid is studied. In experimentally, we used an immiscible drop of 0.08% Xanthan gum in 80:20 glycerol/water falling through 4.5 and 8.3 P Silicon oil. The analytical studies for both interior (viscoelastic drop) and exterior (Newtonian fluid) flows are obtained using the perturbation method. Here, the Oldroyd-B model and Geisekus model are used as the constitutive equation for viscoelastic drop and Newtonian model is used for viscose fluid. For a small enough falling drop, surface tension is dominate force and the shape is spherical. As the volume of the drop increased, the drop falls faster and the free surface deforms from its spherical shape in response to the viscoelastic stresses forces. At a critical drop volume, the free surface develops an inward dimple at its rear stagnation point. The deformation we observe is not due to inertia effect because, the motion of the drop falling is creeping and inertia is negligible. Analytical solution in estimating the Terminal velocity and drop shape have more adaption with experimental Results. In this research, effect of parameters such as Deborah number, Capillary number, viscosity ratio and mobility factor on the motion and shape of the falling drop has been studied analytically. According to this study, by increasing the Deborah number the dimple at the rear end of the drop develop and increase also the Terminal velocity decrease.