We have analysed existence and uniqueness of solutions and continuous dependence on the initial condition (well-posedness) of reset control systems with delay. We have elaborated sufficient conditions, using a concept of well-posedness of reset instants, for existence and uniqueness of solutions of reset control systems with input delay. Moreover, we have extended the results of continuous dependence for delay differential equations to discontinuous initial conditions. Finally, we have used the Hausdorff distance between nominal and perturbed trajectories to develop a concept of continuous dependence, and we have used the previous result to obtain a sufficient condition for continuous dependence on the initial condition for reset control systems with delay.