  We study generalizations of Lorentzian warped products with one-dimensional
base of the form $I\times_f X$, where $I$ is an interval, $X$ is a length space
and $f$ is a positive continuous function. These generalized cones furnish an
important class of Lorentzian length spaces in the sense of [Kunzinger,
S\"amann; Ann. Glob. Anal. Geom. 54(3):399--447, 2018], displaying optimal
causality properties that allow for explicit descriptions of all underlying
notions. In addition, synthetic sectional curvature bounds of generalized cones
are directly related to metric curvature bounds of the fiber $X$. The interest
in such spaces comes both from metric geometry and from General Relativity,
where warped products underlie important cosmological models (FLRW spacetimes).
Moreover, we prove singularity theorems for these spaces, showing that
non-positive lower timelike curvature bounds imply the existence of incomplete
timelike geodesics.
