This work describes the direct observation of deforming water droplets in castor oil, undergoing combined shear and extensional flows in a transparent horizontal annular Couette flow cell, over a wide range of straining fields. This study is motivated by the recognition that many mixing devices of practical importance rely for their effective function on a complex, strongly time-dependent combination of these fundamental straining fields and interactions of the fluid with the confining surfaces. Photographic studies of droplet deformation have identified different modes of breakup and complex droplet structures. This study emphasises the significance of droplet size, constraining geometry and flow history on the dynamic response of droplets within shear and extensional flows.