The effects of shear and elongation on drop deformation are examined through numerical simulation and: experiment. A two-dimensional formulation within the scope of the boundary element method (BEM) is proposed for a drop moving under the influence of an ambient flow inside a channel of a general shape, with emphasis on a convergent-divergent channel. Both the drop and the suspending fluid can be either Newtonian or viscoelastic of the Maxwell type. The predicted planar deformation is found-to provide accurate description of the physical reality, For example, small drops, flowing on the axis, elongate in the convergent part of the channel, then regain their circular form in the divergent part, confirming the experimental observations. Drops placed off-axis are found to rotate during the flow. These drops thus have longer residence time as well as larger and irreversible deformation than those moving on the axis. Both theory and experiment show a difference in deformability for Newtonian and viscoelastic drops in a slit flow. Initially, a Newtonian drop is reluctant to deform, but then deformation is rapid. A viscoelastic drop initially deforms readily, but then the deformation slows down. The slit flow does not flatten drops whose diameter is at least 10 times smaller than the slit gap. The effects of shear and elongation stress, the viscosity ratio, the drop diameter-to-channel-gap ratio, the initial drop position, the interfacial tension, and elasticity of the dispersed and ambient phases were examined using the BEM.