In recent years, artificial neural networks have become the flagship algorithm of artificial intelligence1. In these systems, neuron activation functions are static, and computing is achieved through standard arithmetic operations. By contrast, a prominent branch of neuroinspired computing embraces the dynamical nature of the brain and proposes to endow each component of a neural network with dynamical functionality, such as oscillations, and to rely on emergent physical phenomena, such as synchronization2-6, for solving complex problems with small networks7-11. This approach is especially interesting for hardware implementations, because emerging nanoelectronic devices can provide compact and energy-efficient nonlinear auto-oscillators that mimic the periodic spiking activity of biological neurons12-16. The dynamical couplings between oscillators can then be used to mediate the synaptic communication between the artificial neurons. One challenge for using nanodevices in this way is to achieve learning, which requires fine control and tuning of their coupled oscillations17; the dynamical features of nanodevices can be difficult to control and prone to noise and variability18. Here we show that the outstanding tunability of spintronic nano-oscillators-that is, the possibility of accurately controlling their frequency across a wide range, through electrical current and magnetic field-can be used to address this challenge. We successfully train a hardware network of four spin-torque nano-oscillators to recognize spoken vowels by tuning their frequencies according to an automatic real-time learning rule. We show that the high experimental recognition rates stem from the ability of these oscillators to synchronize. Our results demonstrate that non-trivial pattern classification tasks can be achieved with small hardware neural networks by endowing them with nonlinear dynamical features such as oscillations and synchronization.