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KEY WORDS: Lyapunov Functions, almost-periodic systems, asymptotic stability. For the autonomous system & = f(x), f(0) = 0, the following result strengthening the Lyapunov theorem on asymptotic stability is known. A sufficient condition for asymptotic stability of the zero equflibril,m position is the existence of a positive definite Lyapunov function v(x) such that ~ < 0 and the level surfaces v = const > 0 do not contain entire trajectories [1]. In this paper, we show that this result can be extended (with obvious modifications of the statement) to nonautonomous systems, assllrn~ng that the Lyapunov function v(x) and its partial derivatives are ~]most-periodic in t.Earlier [2], we have shown that in the special case in which an almost-periodic system is linear and v = (G(t)x, z) with ~.lmost-periodic G and G, a certain condition stated below is sufficient for exponential stability (here and in the sequel (x, y> = ~ xiyi).