2011-09-062011-09-062011-09-06http://hdl.handle.net/10222/14164Scalar curvature invariants are scalars formed by the contraction of the Riemann tensor and its covariant derivatives. The main motivation for studying scalar curvature invariants is that they can be used to classify certain spaces uniquely. For example, the I-non-degenerate spaces in Lorentzian signature. Lorentzian metrics that fail to be I-non-degenerate are Kundt metrics and have a very special structure. In this thesis we study pseudo-Riemannian spaces with the property that all of their scalar curvature invariants vanish (VSI spaces) or are constant (CSI spaces). These spaces include pseudo-Riemannian Kundt metrics. VSI and CSI spaces are not only of interest from a mathematical standpoint but also have applications to current theoretical physics. In particular, we focus on studying VSI and CSI spaces in four-dimensional neutral signature. We construct new, very general, classes of CSI and VSI pseudo-Riemannian spaces.en