This paper deals with the problem of stability for systems with delay varying in an interval. A new Lyapunov functional, which makes use of the information of both the lower and upper bounds of the interval time-varying delay, is proposed to derive some new stability criteria. Furthermore, the relationship of the time-varying delay and its lower bound and upper bound is taken into account. As a result, some less conservative delay-dependent stability criteria are obtained without ignoring any useful information in the derivative of Lyapunov functional, which are established in the forms of linear matrix inequalities. Numerical examples are provided to show that the obtained results are better than existing ones.

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