This paper focuses on the dynamic behavior of fractional-order inertial Cohen-Grossberg neural networks with time-varying delays. Using Riemann-Liouville fractional calculus properties and initial value conditions, we divide the definition field $[0, + \infty)$ of $t$ into two intervals according to the time-varying delays $\tau_{ij}(t)$ of the system: $[0,\tau_{ij}(t)]$ and $[\tau_{ij}(t),+\infty)$, and then we deduce the relationship between the fractional integrals of the state function $x_{i}(t-\tau_{ij}(t))$ when $t$ is in $[0,\tau_{ij}(t)]$ and $[\tau_{ij}(t),+\infty)$. By introducing the Mittag-Leffler function, with the help of finite increment formula of Lagrange mean-value theorem and Arzela-Ascoli theorem that when the function sequence is equi-continuous and uniform, there is a uniformly convergent subsequence, we get the sufficient conditions to determine the global Mittag-Leffler stability and global asymptotic $\omega$-periodicity. Finally, we give numerical simulation examples to verify the effectiveness of the theoretical results.