In §8.1, we discuss the problem of defining singularities in spacetime. We adopt b-incompleteness as an indication that singular points have been cut out of spacetime, and characterize two ways in which b-incompleteness can be associated with some form of curvature singularity. In §8.2, four theorems are given to prove the existence of incompleteness under a wide variety of situations. In §8.3 we give Schmidt’s construction of the b-boundary which represents the singular points of spacetime. In §8.4 we prove that the singularities predicted by at least one of the the theorems cannot be just a discontinuity in the curvature tensor. We also show that there is not only one incomplete geodesic, but a three-parameter family of them. In §8.5 we discuss the situation in which the incomplete curves are totally or partially imprisoned in a compact region of spacetime, shown to be related to non-Hausdorff behaviour of the b-boundary. We show that in a generic spacetime, an observer travelling on one of these incomplete curves would experience infinite curvature forces. We also show that the kind of behaviour which occurs in Taub–NUT space cannot happen if there is some matter present.