Smooth Lorentzian warped products with one-dimensional base are important examples of spacetimes: They contain well-known physical models (such as the FLRW spacetimes) and admit a very simple description of causal curves and geodesics. We will examine what happens if one replaces the Riemannian manifold in the fiber with a locally compact length space. As long as the warping function is continuous and positive there still exists a natural notion of causal curves and their length and hence also of the causality relations on the product. This turns such ""generalized cones"" into a Lorentzian length space. Analogous to the smooth case the causal structure of such spaces is very simple and one has an explicit description of the causal relations. Inspired by the well-developed Riemannian theory of warped products of length spaces, we obtain some results concerning timelike curvature bounds and a singularity theorem in this setting. This talk is based on joint work with S. B. Alexander, M. Kunzinger and C. Sämann.