Stability of a set of systems with norm bounded nonlinear terms and arbitrary time-varying as well as distributed delays is studied. A novel approach to this problem, based on deriving bounds for the norms of system solutions, is developed. A sharp estimate for the maximal Lyapunov exponent of the solutions, expressed in the bounds for the uncertain parameters, is found. The subsystems, for which the obtained estimate is attained, are indicated. Using these results, a delay-independent necessary and sufficient stability condition for the considered set of systems is derived. For a system with prescribed parameters, sufficient conditions for exponential stability and upper bound for the maximal Lyapunov exponent are obtained. The proposed approach is applied to illustrative examples which contrast its efficiency.