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Randomized benchmarking (RB) protocols are the most widely used methods for assessing the performance of quantum gates. However, the existing RB methods either do not scale to many qubits or cannot benchmark a universal gate set. Here, we introduce and demonstrate a technique for scalable RB of many universal and continuously parameterized gate sets, using a class of circuits called randomized mirror circuits. Our technique can be applied to a gate set containing an entangling Clifford gate and the set of arbitrary single-qubit gates, as well as gate sets containing controlled rotations about the Pauli axes. We use our technique to benchmark universal gate sets on four qubits of the Advanced Quantum Testbed, including a gate set containing a controlled-S gate and its inverse, and we investigate how the observed error rate is impacted by the inclusion of non-Clifford gates. Finally, we demonstrate that our technique scales to many qubits with experiments on a 27-qubit IBM Q processor. We use our technique to quantify the impact of crosstalk on this 27-qubit device, and we find that it contributes approximately 2/3 of the total error per gate in random many-qubit circuit layers.