We investigate the problem of constructing spanners
for a given set of points that are tolerant for edge/vertex faults.
Let *S* be a set of *n* points in the *d*-dimensional space
and let *k* be an integer number.
A *k*-edge/vertex fault tolerant spanner for *S* has the property that after
the deletion of *k* arbitrary edges/vertices each pair of points in the
remaining graph is still connected by a short path.

Recently it was shown that for each set *S* of *n* points there exists
a *k*-edge/vertex fault tolerant spanner with
*O*(*k*^{2}*n*) edges which
can be constructed in
*O*(*n* log *n* + *k*^{2}*n*) time.
Furthermore, it was shown that for each set *S* of *n* points there exists
a *k*-edge/vertex fault tolerant spanner whose degree is bouned by
*O*(*c*^{k+1}) for some constant *c*.

Our first contribution is a construction of a *k*-vertex fault tolerant
spanner with *O*(*kn*) edges which is a tight bound.
The computation takes
*O*(*n* log^{d-1}*n* + *k n *loglog *n*) time.
Then we show that the same k-vertex fault tolerant spanner is also
*k*-edge fault tolerant.
Thereafter, we construct a k-vertex fault tolerant spanner with
*O*(*k*^{2}*n*)
edges whose degree is bounded by *O*(*k*^{2}).
Finally, we give a more natural but stronger
definition of *k*-edge fault tolerance which not necessarily can be satisfied
if one allows only simple edges between the points of *S*. We investigate the
question whether Steiner points help. We answer this question affirmatively
and prove *Theta*(*kn*) bounds on the number of Steiner points and on the
number of edges in such spanners.