  In this paper we investigate four dimensional Lorentzian spacetimes with
constant curvature invariants ($CSI$ spacetimes). We prove that if a four
dimensional spacetime is $CSI$, then either the spacetime is locally
homogeneous or the spacetime is a Kundt spacetime for which there exists a
frame such that the positive boost weight components of all curvature tensors
vanish and the boost weight zero components are all constant. We discuss some
of the properties of the Kundt-$CSI$ spacetimes and their applications. In
particular, we discuss $\mathcal{I}$-symmetric spaces and degenerate Kundt
$CSI$ spacetimes.
