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 $ abla(Riem), abla^{(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.
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