The authors consider scalar delay differential equations \[ x'(t) = -\delta x(t)+f(t,x_t)\tag{1} \] with nonlinear \(f\) satisfying a sort of negative feedback condition combined with a boundedness condition. The well-known Mackey-Glass-type equations, equations satisfying the Yorke condition, and equations with maxima are special cases of (1). Here, a criterion is established for the global asymptotical stability of a unique steady state to (1). As an example, Nicholson’s blowflies equation is studied where the computations support the Smith conjecture about the equivalence between global and local asymptotical stabilities in this population model.