We present the results of an experimental investigation of drop deformation and breakup in pressure-driven flow through a cylindrical capillary, where the drop and/or the suspending fluid consist of a viscoelastic polymer solution. Experimental observations of drop shape are reported for various concentrations of the polymer solution in order to examine the role of elasticity on drop dynamics. As the strength of the imposed flow increases, large drops eventually become unstable and break up. Fluid elasticity has a significant effect on the breakup behavior of drops, and different modes of drop breakup are observed depending on the polymer concentration in the interior and exterior phases. The various modes of drop breakup observed in the experiments, as well as the critical conditions for the onset of each mode of breakup, are reported. These include the transient stretching and re-entrant jet mechanisms previously reported for Newtonian two-phase systems, as well as tip-streaming of the interior fluid from a trailing cusp, tip-streaming of the exterior fluid within the drop from a re-entrant cusp, and rim-streaming which is characterized by a stream of small droplets being continuously ejected from several points along the trailing rim of the drop (on or near the rear stagnation ring). Tip-streaming of the interior fluid from a trailing cusp occurs when the exterior fluid is viscoelastic, while tip-streaming of the exterior fluid from a re-entrant cusp occurs when the interior fluid is viscoelastic with moderate polymer concentration, and rim-streaming occurs when the interior fluid is viscoelastic with high polymer concentration. In all cases, the critical capillary number for the onset of drop breakup is a decreasing function of drop size.