The paper discusses the stability and asymptotic behavior of fractional-order differential equations involving both delayed as well as nondelayed terms. As the main results, the necessary and sufficient conditions guaranteeing asymptotic stability of its zero solution are presented, including asymptotic formulae for all its solutions. Since this equation represents a basic test equation for numerical analysis of delay differential equations of fractional type, the knowledge of its optimal stability conditions is crucial for investigations of numerical stability. Theoretical conclusions are supported by comments and comparisons distinguishing behaviour of a fractional-order delay equation from its integer-order pattern.