We show that for each gamma in (0,2), there is a unique metric (i.e., distance function) associated with gamma -Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) h, there is a unique random metric D_h associated with the Riemannian metric tensor “e^{gamma h} (dx^2 + dy^2)” on {mathbb {C}} which is characterized by a certain list of axioms: it is locally determined by h and it transforms appropriately when either adding a continuous function to h or applying a conformal automorphism of mathbb {C} (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The gamma -LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding et al. (Tightness of Liouville first passage percolation for gamma in (0,2), 2019. arXiv:1904.08021) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when gamma = sqrt{8/3}, our metric coincides with the sqrt{8/3}-LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general gamma in (0,2), we conjecture that our metric is the Gromov–Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems.