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Implementations of artificial neural networks that borrow analogue techniques could potentially offer low-power alternatives to fully digital approaches1–3. One notable example is in-memory computing based on crossbar arrays of non-volatile memories4–7that execute, in an analogue manner, multiply–accumulate operations prevalent in artificial neural networks. Various non-volatile memories—including resistive memory8–13, phase-change memory14,15and flash memory16–19—have been used for such approaches. However, it remains challenging to develop a crossbar array of spin-transfer-torque magnetoresistive random-access memory (MRAM)20–22,  despite the technology’s practical advantages such as endurance and large-scale commercialization5. The difficulty stems from the low resistance of MRAM, which would result in large power consumption in a conventional crossbar array that uses current summation for analogue multiply–accumulate operations. Here we report a 64 × 64 crossbar array based on MRAM cells that overcomes the low-resistance issue with an architecture that uses resistance summation for analogue multiply–accumulate operations. The array is integrated with readout electronics in 28-nanometre complementary metal–oxide–semiconductor technology. Using this array, a two-layer perceptron is implemented to classify 10,000 Modified National Institute of Standards and Technology digits with an accuracy of 93.23 per cent (software baseline: 95.24 per cent). In an emulation of a deeper, eight-layer Visual Geometry Group-8 neural network with measured errors, the classification accuracy improves to 98.86 per cent (software baseline: 99.28 per cent). We also use the array to implement a single layer in a ten-layer neural network to realize face detection with an accuracy of 93.4 per cent.