This chapter is a brief survey of the recent results of the author devoted to the stability of linear functional-differential equations in a Banach space. We consider autonomous and non-autonomous equations with bounded and unbounded operators. The illustrative examples with partial differential and integro-differential equations are also presented. These examples show that the obtained stability conditions allow us to avoid in appropriate situations the construction of the Krasovskij-Lyapunov type functionals. In the case of unbounded operators we generalize the well-known Dyson-Phillips theorem on perturbations of strongly continuous semigroups to the fundamental solutions of differential-difference equations in a Banach space.