The paper is devoted to a class of functional differential equations in a Banach space. Using Razumikhin's technique, the authors establish several criteria on uniform asymptotic stability in terms of two measures. In particular, a first order scalar integral-differential equation with unbounded delay is considered. Under some restrictions on the coefficients, explicit stability conditions are derived. In addition, an equation in a Banach space with a separated unbounded linear operator and a Lipschitz continuous nonlinearity is investigated. It is assumed that the linear operator generates an exponentially stable semigroup. Moreover, the authors establish stability conditions for a linear time-variant parabolic equation with unbounded delay.