  A {\em 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 {\em 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 \emph{additive} spanner for (the surviving part of) $G$, satisfying
$dist(s,t,H\setminus \{v\}) \leq dist(s,t,G\setminus \{v\})+\beta$ for every
$s,t,v \in V$. Recently, the problem of constructing fault-tolerant additive
spanners resilient to the failure of up to $f$ \emph{edges} has been considered
by Braunschvig et. al. The problem of handling \emph{vertex} failures was left
open therein. In this paper we develop new techniques for constructing additive
FT-spanners overcoming the failure of a single vertex in the graph. Our first
result is an FT-spanner with additive stretch $2$ and $\widetilde{O}(n^{5/3})$
edges. Our second result is an FT-spanner with additive stretch $6$ and
$\widetilde{O}(n^{3/2})$ edges. The construction algorithm consists of two main
components: (a) constructing an FT-clustering graph and (b) applying a modified
path-buying procedure suitably adopted to failure prone settings. Finally, we
also describe two constructions for {\em fault-tolerant multi-source additive
spanners}, aiming to guarantee a bounded additive stretch following a vertex
failure, for every pair of vertices in $S \times V$ for a given subset of
sources $S\subseteq V$. The additive stretch bounds of our constructions are 4
and 8 (using a different number of edges).
