Non-classical states of light, such as entangled photon pairs and number states, are essential for fundamental tests of quantum mechanics and optical quantum technologies. The most widespread technique for creating these quantum resources is spontaneous parametric down-conversion of laser light into photon pairs(1). Conservation of energy and momentum in this process, known as phase-matching, gives rise to strong correlations that are used to produce two-photon entanglement in various degrees of freedom(2-9). It has been a longstanding goal in quantum optics to realize a source that can produce analogous correlations in photon triplets, but of the many approaches considered, none has been technically feasible(10-17). Here we report the observation of photon triplets generated by cascaded down-conversion. Each triplet originates from a single pump photon, and therefore quantum correlations will extend over all three photons(18) in a way not achievable with independently created photon pairs(19). Our photon-triplet source will allow experimental interrogation of novel quantum correlations(20), the generation of tripartite entanglement(12,21) without post-selection and the generation of heralded entangled photon pairs suitable for linear optical quantum computing(22). Two of the triplet photons have a wavelength matched for optimal transmission in optical fibres, suitable for three-party quantum communication(23). Furthermore, our results open interesting regimes of non-linear optics, as we observe spontaneous down-conversion pumped by single photons, an interaction also highly relevant to optical quantum computing.