In this paper we investigate the left invertibility problem for a class of nonlinear time-delay systems. In both cases of time delay systems with and without internal dynamics the invertibility conditions are given. A new approach based on the use of higher order sliding mode observer is developed for finite-time left invertibility and for asymptotic left inversion. Causal and non-causal estimation of the unknown inputs are respectively discussed. The results are illustrated by numerical examples in order to show the efficiency of the method and its limits.