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This paper mainly examines the stability and the existence of Hopf bifurcation of fractional-order BAM neural networks with multiple delays. Based on Laplace transform, stability criterion and Hopf bifurcation theory of fractional-order differential equations, a new sufficient criterion to guarantee the stability and the existence of Hopf bifurcation for the involved fractional-order BAM neural networks with multiple delays is established. The investigation manifests that time delay plays an important role in maintaining network stability and controlling the appearance of Hopf bifurcation of fractional-order BAM neural networks. Numerical simulations are performed to check the rationality of the analytical conclusions. The obtained theoretical predictions of this paper have extremely vital guiding significance in designing and controlling networks. • The stability and the existence of Hopf bifurcation of fractional-order BAM neural networks with multiple delays is investigated. • There are few works on bifurcation of fractional delayed high-dimensional neural networks. • Up to now, the topic on Hopf bifurcation of fractional order high-dimensional neural networks with multiple delays is very rare. [ABSTRACT FROM AUTHOR]