Liouville quantum gravity (LQG) is a natural model for a two-dimensional continuum random geometry. It originated from work on string theory and conformal field theory in the 1980s. In the past decade, LQG has been rigorously understood as a random measure on a two-dimensional surface, by taking a limit of measures on suitable smooth approximations. However, only at a single special temperature has a metric space structure for LQG been constructed. I will discuss recent work on the tightness of a sequence of natural discretized LQG metrics, the subsequential limits of which thus form natural candidates for a continuum metric for LQG . This is joint work with Jian Ding.