In this paper, we develop the Lyapunov-Razumikhin method to finite-time stability (FTS) and finitetime contractive stability (FTCS) of time-delay systems. Several Lyapunov-based sufficient conditions for establishing these FTS properties are obtained. Then the theoretical results are applied to FTS and FTCS for a class of linear time-varying (LTV) time-delay system. The efficiency of the proposed criteria is illustrated by three numerical examples, where a stabilizing memoryless controller for FTCS of a second-order LTV system with time delay is proposed. (C) 2019 Elsevier Ltd. All rights reserved.