  Kundt spacetimes are of great importance in general relativity in 4
dimensions and have a number of topical applications in higher dimensions in
the context of string theory. The degenerate Kundt spacetimes have many special
and unique mathematical properties, including their invariant curvature
structure and their holonomy structure. We provide a rigorous geometrical
kinematical definition of the general Kundt spacetime in 4 dimensions;
essentially a Kundt spacetime is defined as one admitting a null vector that is
geodesic, expansion-free, shear-free and twist-free. A Kundt spacetime is said
to be degenerate if the preferred kinematic and curvature null frames are all
aligned. The degenerate Kundt spacetimes are the only spacetimes in 4
dimensions that are not $\mathcal{I}$-non-degenerate, so that they are not
determined by their scalar polynomial curvature invariants. We first discuss
the non-aligned Kundt spacetimes, and then turn our attention to the degenerate
Kundt spacetimes. The degenerate Kundt spacetimes are classified algebraically
by the Riemann tensor and its covariant derivatives in the aligned kinematic
frame; as an example, we classify Riemann type D degenerate Kundt spacetimes in
which $\nabla(Riem),\nabla^{(2)}(Riem)$ are also of type D. We discuss other
local characteristics of the degenerate Kundt spacetimes. Finally, we discuss
degenerate Kundt spacetimes in higher dimensions.
