  Physics considerations suggest that a theory of Liouville quantum gravity
(LQG) should exist for all values of matter central charge $\mathbf{c} \in
(-\infty,25)$. Probabilists have rigorously defined LQG as a random metric
measure space for $\mathbf{c} < 1$; however, they have only very recently begun
to explore LQG in the $\mathbf{c} \in (1,25)$ phase. We define a random metric
associated to LQG for all $\mathbf{c} < 25$ by a collection of axioms analogous
to axioms stated in the $\mathbf{c} < 1$ setting. We show that such a metric
exists for each $\mathbf{c}$ by considering an approximating metric known as
Liouville first passage percolation. Ding and Gwynne proved that these
approximating metrics are tight in a suitably chosen topology; we show that
every subsequential limit satisfies our axioms. In particular, our result
defines a metric associated to LQG in the critical case $\mathbf{c} =1$.
  The metrics we define for $\mathbf{c} \in (1,25)$ exhibit geometric behavior
that sharply contrasts with the $\mathbf{c} < 1$ regime. We show that, for
$\mathbf{c} \in (1,25)$, the metrics do not induce the Euclidean topology since
they a.s. have a dense (measure zero) set of singular points, points at
infinite distance from all other points. We use this fact to prove that a.s.
the metric ball is not compact and its boundary has infinite Hausdorff
dimension. Despite these differences, we demonstrate that many properties of
LQG metrics for $\mathbf{c} < 1$ extend in some form to the entire range
$(-\infty, 25)$. We show that the metrics are a.s. reverse H\"older continuous
with respect to the Euclidean metric, are a.s. complete and geodesic away from
the set of singular points, and satisfy the bounds for set-to-set distances
that hold in the $\mathbf{c} < 1$ phase. Finally, we prove that the metrics
satisfy a version of the (geometric) Knizhnik-Polyakov-Zamolodchikov (KPZ)
formula.
