The attachment pads of some beetles, spiders, flies, and geckos are covered by a dense array of long hairs with characteristic geometries. This curious surface topography allows them to firmly attach to and easily release from almost any kind of surface. In a technological context, such reversible adhesion could enable robots to walk along walls or ceilings, or lead to new medical devices, disposable plasters, reusable adhesive tapes, etc. Artificial fibrillar surfaces mimicking nature's design have been recently fabricated, but their adhesion performance is still far from that of natural systems. This review describes the progress in this field during the last few years and discusses the issues pending for the future.