Extrusion-based additive manufacturing (AM) has recently become widespread for the layer-by-layer fabrication of three-dimensional prototypes and components even with highly complex shapes. This technology involves extrusion through a nozzle by means of a plunger-, filament- or screw-based mechanism; where necessary, this is preceded by heating of the feedstock material to reduce its viscosity sufficiently to facilitate extrusion. Extrusion-based AM offers greater design freedom, larger building volumes and more cost-efficient production than liquid- and powder-based AM processes. Although this technology was originally developed for polymeric filament materials, it is now increasingly applied to a wide variety of material classes, including metallic, edible and construction materials. This is in part thanks to the recent development of AM-specific feedstock materials (AM materials), in which materials that are not intrinsically suited to extrusion, for example because of high melting points or brittleness, are combined with other, usually polymeric materials that can be more readily extruded. This paper comprehensively and systematically reviews the state of the art in the field of extrusion-based AM, including the techniques applied and the individual challenges and developments in each materials class for which the technology is being developed. The paper includes material- and process-centred suitability analysis of extrusion-based AM, and a comparison of this technology with liquid- and powder-based AM processes. Prospective applications of this technology are also briefly discussed.