This paper surveys the research literature on robust tactile and haptic illusions. The illusions are organized into two categories. The first category relates to objects and their properties, and is further differentiated in terms of haptic processing of material versus geometric object properties. The second category relates to haptic space, and is further differentiated in terms of the observer's own body versus external space. The illusions are initially described and where possible addressed in terms of their functional properties and/or underlying neural processes. The significance of these illusions for the design of tactile and haptic displays is also discussed. We conclude by briefly considering a number of important general themes that have emerged in the materials surveyed.