Photonic crystals are attracting current interest for a variety of reasons, such as their ability to inhibit the spontaneous emission of light(1,2). This and related properties arise from the formation of photonic bandgaps, whereby multiple scattering of photons by lattices of periodically varying refractive indices acts to prevent the propagation of electromagnetic waves having certain wavelengths. One route to forming photonic crystals is to etch two-dimensional periodic lattices of vertical air holes into dielectric slab waveguides(3-7). Such structures can show complete photonic bandgaps(8-10), but only for large-diameter air holes in materials of high refractive index (such as gallium arsenide, n = 3.69), which unfortunately leads to significantly reduced optical transmission when combined with optical fibres of low refractive index. It has been suggested that quasicrystalline (rather than periodic) lattices can also possess photonic bandgaps(11-14). Here we demonstrate this concept experimentally and show that it enables complete photonic bandgaps-non-directional and for any polarization-to be realized with small air holes in silicon nitride (n = 2.02), and even glass (n = 1.45). These properties make photonic quasicrystals promising for application in a range of optical devices(14-18).