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This paper focuses on the quasi-invariant set (QIS), global attracting set (GAS) and global exponential attracting set (GEAS) of competitive neural networks (CNNs) with time-varying and infinite distributed delays. For these purposes, based on the characteristics of nonnegative matrix and M -matrix, a new bidirectional delay integral inequality and a novel integro-differential inequality are first established. From the founded integral inequality, the existence conditions of the QIS and the GAS of the discussed system are obtained. Besides, the existence conditions of the GEAS are also given by the proposed integro-differential inequality, which gets rid of the construction of complex Lyapunov functions and functionals. The frameworks of the QIS, GAS and GEAS are also given. A numerical example is analyzed to confirm the validity of the obtained results in the end. • The quasi-invariant, global attracting and global exponential attracting sets for competitive neural networks with mixed time-varying delays are studied. • Some new integral and integro-differential inequalities are established based on the characteristics of nonnegative matrix and M-matrix. • Some effective criteria for the quasi-invariant, global attracting and global exponential attracting sets are established. • The global exponential attracting set is straightway obtained by the integro-differential inequality, which gets rid of the construction of complex Lyapunov functions and functionals. [ABSTRACT FROM AUTHOR]