A fault-tolerant structure for a network is required to continue functioning following the failure of some of the network's edges or vertices. In this paper, we address the problem of designing a fault-tolerant additive spanner, namely, a subgraph H of the network G such that subsequent to the failure of a single vertex, the surviving part of H still contains an additive spanner for (the surviving partthe problem of constructing fault-tolerant additive spanners resilient to the failure of up to f edges has been considered (Braunschvig et al. Proceedings of the WG, pp 206-214, 2012). The problem of handling vertex failures was left open therein. In this paper we develop new techniques for constructing additive FTspanners overcoming the failure of a single vertex in the graph.Our first result is an FT-spanner with additive stretch 2 and O(n 5/3 ) edges. Our second result is an FT-spanner with additive stretch 6 and O(n 3/2 ) edges. The construction algorithm consists of two main components: (a) constructing an FT-clustering graph and (b) applying a modified path-buying procedure suitably adapted to failure prone settings. Finally, we also describe two constructions for fault-tolerant multisource additive spanners, aiming to guarantee a bounded additive stretch following a vertex failure, for every pair of vertices in S × V for a given subset of sources S ⊆ V . The additive stretch bounds of our constructions are 4 and 8 (using a different number of edges).