This paper investigates the finite-time stability (FTS) and stabilization for a class of nonlinear time-delay systems and proposes some new results. First, an equivalent form is presented for the systems by a coordinate transformation, and a criterion on the FTS is obtained for nonlinear time-delay systems via the Razumikhin approach (R-approach). Then, based on the criterion and the equivalent form, some delay-independent and delay-dependent conditions on the FTS are derived for this class of systems by constructing a new Lyapunov function. Third, we use the obtained FTS results to investigate the finite-time stabilization problem and present a control design procedure for the time-delay systems. It is shown that our method can avoid solving nonlinear matrix inequalities by transferring them into the form of linear matrix inequalities, which can be easily solved with the LMI Toolbox of MATLAB. Finally, study of illustrative examples shows that the results obtained in this paper work very well in the FTS analysis and control design for the nonlinear time-delay systems.