Fibrillar structures are found on the attachment pads of insects and small reptiles. These structures enable exquisite conformation to rough surfaces, increase the number of van der Waals interactions between the structure and the target surface, and thus enhance adhesion. Biomimetic adhesives replicate this effect by patterning polymer films with micron- or sub-micron-sized protrusions. Numerical contact-mechanics models as well as experimental adhesion measurements have been reported for a variety of protrusion shapes including flat, rounded, mushroom and spatula geometries. Although superior adhesion has been reported for the mushroom and spatula tip geometries, straight, flat-tipped pillars offer the potential for much simpler mass production such as by injection moulding and are thus the focus of this review. Existing models for straight, flat-tipped pillar arrays do not fully agree with reported experimental results. Analytical models are generally limited to elastic materials, and inherently assume that neighbouring pillars behave independently. For elastic pillars in close proximity, however, pillars do in fact interact mechanically, affecting adhesion. Moreover, visco- and hyper-elastic materials are often used in practice, yet dissipative effects receive little attention in analytical models. We find that no study has conclusively investigated the limit of adhesive strength achievable by fibrillar adhesives. It also remains unclear what happens to the adhesive strength as the areal density of contacting regions approaches 100%.