In this paper, the stability problem for time-varying nonlinear delayed systems is investigated. An infinite-dimensional version of the nested Matrosov function theorem is proposed to conclude uniform global asymptotic stability for time-varying nonlinear delayed systems. Such a stability theorem requires a weak Lyapunov functional and some auxiliary functionals and the upper bounds of the derivatives on these functionals must satisfy some nested conditions. The proposed results can be used to analyze uniform asymptotic stability of time-varying delayed systems where LaSalle's invariance principle cannot be applied and Barbalat's lemma is not sufficient to conclude the uniform attractivity property. An example is given to illustrate the effectiveness of the proposed stability theorem. (C) 2019 Elsevier Ltd. All rights reserved.