We present a survey of recent work on robot manipulation and sensing of deformable objects, a field with relevant applications in diverse industries such as medical (e.g. surgery assistance), food handling, manufacturing, and domestic chores (e.g. folding clothes). We classify the reviewed approaches into four categories based on the type of object they manipulate. Furthermore, within this object classification we divide the approaches based on the particular task they perform on the deformable object. Finally, we conclude this survey with a discussion of the current state of the art and propose future directions within the proposed classification.