  Inspired by some Lorentzian versions of the notion of metric and length space
introduced by Kunzinger and S\"amman, and more recently, by M\"uller, and
Minguzzi and S\"uhr, we revisit the notion of Lorentzian metric space in order
to later construct the c-completion of these general objects. We not only prove
that this construction is feasible in great generality for these objects,
including spacetimes of low regularity, but also endow the c-completion with a
structure of Lorentzian metric space by itself. We also prove that the
c-completion constitutes a well-suited extension of the original space, which
really completes it in a precise sense and becomes sensible to certain causal
properties of that space.
