Entangled photon pairs are an important resource in quantum optics(1), and are essential for quantum information(2) applications such as quantum key distribution(3,4) and controlled quantum logic operations(5). The radiative decay of biexcitons - that is, states consisting of two bound electron - hole pairs - in a quantum dot has been proposed as a source of triggered polarization-entangled photon pairs(6). To date, however, experiments have indicated that a splitting of the intermediate exciton energy yields only classically correlated emission(7-9). Here we demonstrate triggered photon pair emission from single quantum dots suggestive of polarization entanglement. We achieve this by tuning the splitting to zero, through either application of an in-plane magnetic field or careful control of growth conditions. Entangled photon pairs generated 'on demand' have significant fundamental advantages over other schemes(10-13), which can suffer from multiple pair emission, or require post-selection techniques or the use of photon-number discriminating detectors. Furthermore, control over the pair generation time is essential for scaling many quantum information schemes beyond a few gates. Our results suggest that a triggered entangled photon pair source could be implemented by a simple semiconductor light-emitting diode(14).