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Deep neural networks have achieved remarkable success on a variety of mission-critical tasks. However, recent studies show that deep neural networks are vulnerable to backdoor attacks, where the attacker releases backdoored models that behave normally on benign samples but misclassify any trigger-imposed samples to a target label. Unlike adversarial examples, backdoor attacks manipulate both the inputs and the model, perturbing samples with the trigger and injecting backdoors into the model. In this paper, we propose a novel attention-based evasive backdoor attack, dubbed ATTEQ-NN. Different from existing works that arbitrarily set the trigger mask, we carefully design an attention-based trigger mask determination framework, which places the trigger at the crucial region with the most significant influence on the prediction results. To make the trigger-imposed samples appear more natural and imperceptible to human inspectors, we introduce a Quality-of-Experience (QoE) term into the loss function of trigger generation and carefully adjust the transparency of the trigger. During the process of iteratively optimizing the trigger generation and the backdoor injection components, we propose an alternating retraining strategy, which is shown to be effective in improving the clean data accuracy and evading some model-based defense approaches. We evaluate ATTEQ-NN with extensive experiments on VGG- Flower, CIFAR-10, GTSRB, and CIFAR-100 datasets. The results show that ATTEQ-NN can increase the attack success rate by as high as 82% over baselines when the poison ratio is low while achieving a high QoE of the backdoored samples. We demonstrate that ATTEQ-NN reaches an attack success rate of more than 41.7% in the physical world under different lighting conditions and shooting angles. ATTEQ-NN preserves an attack success rate of more than 92.5% even if the original backdoored model is fine-tuned with clean data. Our user studies show that the backdoored samples generated by ATTEQ-NN are indiscernible under visual inspections. ATTEQ-NN is shown to be evasive to state-of-the-art defense methods, including model pruning, NAD, STRIP, NC, and MNTD.