  We study Lorentzian spacetimes for which all scalar invariants constructed
from the Riemann tensor and its covariant derivatives are constant ($CSI$
spacetimes). We obtain a number of general results in arbitrary dimensions. We
study and construct warped product $CSI$ spacetimes and higher-dimensional
Kundt $CSI$ spacetimes. We show how these spacetimes can be constructed from
locally homogeneous spaces and $VSI$ spacetimes. The results suggest a number
of conjectures. In particular, it is plausible that for $CSI$ spacetimes that
are not locally homogeneous the Weyl type is $II$, $III$, $N$ or $O$, with any
boost weight zero components being constant. We then consider the
four-dimensional $CSI$ spacetimes in more detail. We show that there are severe
constraints on these spacetimes, and we argue that it is plausible that they
are either locally homogeneous or that the spacetime necessarily belongs to the
Kundt class of $CSI$ spacetimes, all of which are constructed. The
four-dimensional results lend support to the conjectures in higher dimensions.
