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First, we set up in an appropriate way the initial value problem for nonlinear delay differential equations with a Riemann-Liouville (RL) fractional derivative. We define stability in time and generalize Mittag-Leffler stability for RL fractional differential equations and we study stability properties by an appropriate modification of the Razumikhin method. Two different types of derivatives of Lyapunov functions are studied: the RL fractional derivative when the argument of the Lyapunov function is any solution of the studied problem and a special type of Dini fractional derivative among the studied problem. [ABSTRACT FROM AUTHOR]