Randomness is important for many information processing applications, including numerical modelling and cryptography(1,2). Device-independent quantum random-number generation (DIQRNG)(3,4) based on the loophole-free violation of a Bell inequality produces genuine, unpredictable randomness without requiring any assumptions about the inner workings of the devices, and is therefore an ultimate goal in the field of quantum information science(5-7). Previously reported experimental demonstrations of DIQRNG(8,9) were not provably secure against the most general adversaries or did not close the 'locality' loophole of the Bell test. Here we present DIQRNG that is secure against quantum and classical adversaries(10-12). We use state-of-the-art quantum optical technology to create, modulate and detect entangled photon pairs, achieving an efficiency of more than 78 per cent from creation to detection at a distance of about 200 metres that greatly exceeds the threshold for closing the 'detection' loophole of the Bell test. By independently and randomly choosing the base settings for measuring the entangled photon pairs and by ensuring space-like separation between the measurement events, we also satisfy the no-signalling condition and close the 'locality' loophole of the Bell test, thus enabling the realization of the loophole-free violation of a Bell inequality. This, along with a high-voltage, high-repetition-rate Pockels cell modulation set-up, allows us to accumulate sufficient data in the experimental time to extract genuine quantum randomness that is secure against the most general adversaries. By applying a large (137.90 gigabits x 62.469 megabits) Toeplitz-matrix hashing technique, we obtain 6.2469 x 10(7) quantum-certified random bits in 96 hours with a total failure probability (of producing a random number that is not guaranteed to be perfectly secure) of less than 10(-5). Our demonstration is a crucial step towards transforming DIQRNG from a concept to a key aspect of practical applications that require high levels of security and thus genuine randomness(7). Our work may also help to improve our understanding of the origin of randomness from a fundamental perspective.