Title: Universal discriminative quantum neural networks. Abstract: Quantum mechanics fundamentally forbids deterministic discrimination of quantum states and processes. However, the ability to optimally distinguish various classes of quantum data is an important primitive in quantum information science. In this work, we trained near-term quantum circuits to classify data represented by quantum states using the Adam stochastic optimization algorithm. This is achieved by iterative interactions of a classical device with a quantum processor to discover the parameters of an unknown non-unitary quantum circuit. This circuit learns to simulate the unknown structure of a generalized quantum measurement, or positive-operator valued measure (POVM), that is required to optimally distinguish possible distributions of quantum inputs. Notably we used universal circuit topologies, with a theoretically motivated circuit design which guaranteed that our circuits can perform arbitrary input-output mappings. Our numerical simulations showed that quantum circuits could be trained to discriminate among various pure and mixed quantum states, exhibiting a trade-off between minimizing erroneous and inconclusive outcomes with comparable performance to theoretically optimal POVMs. We trained the circuit on different classes of quantum data and evaluated the generalization error on unseen quantum data. This generalization power hence distinguishes our work from standard circuit optimization and provides an example of quantum machine learning for a task that has inherently no classical analogue. 