  In this paper we consider pseudo-Riemannian spaces of arbitrary signature for
which all of their polynomial curvature invariants vanish (VSI spaces). We
discuss an algebraic classification of pseudo-Riemannian spaces in terms of the
boost weight decomposition and define the ${\bf S}_i$- and ${\bf
N}$-properties, and show that if the curvature tensors of the space possess the
${\bf N}$-property then it is a VSI space. We then use this result to construct
a set of metrics that are VSI. All of the VSI spaces constructed possess a
geodesic, expansion-free, shear-free, and twist-free null-congruence. We also
discuss the related Walker metrics.
