  We introduce an analogue of the theory of length spaces into the setting of
Lorentzian geometry and causality theory. The r\^ole of the metric is taken
over by the time separation function, in terms of which all basic notions are
formulated. In this way we recover many fundamental results in greater
generality, while at the same time clarifying the minimal requirements for and
the interdependence of the basic building blocks of the theory. A main focus of
this work is the introduction of synthetic curvature bounds, akin to the theory
of Alexandrov and CAT$(k)$-spaces, based on triangle comparison. Applications
include Lorentzian manifolds with metrics of low regularity, closed cone
structures, and certain approaches to quantum gravity.
