Last and first passage percolation in two dimensions are classical discrete models of random directed planar Euclidean metrics in the KPZ universality class. Their scaling limit is described by the directed landscape of Dauvergne-Ortmann-Virág. 

Random planar maps are classical discrete models of random undirected planar fractal metrics in the LQG universality class. Their scaling limit is described by the (undirected) LQG metric of Ding-Dubédat-Dunlap-Falconet and Gwynne-Miller. 

 We present recent progress on the study of longest and shortest directed paths in the following natural model of directed random planar maps: the uniform infinite bipolar-oriented triangulation (UIBOT), which is the local limit of uniform bipolar-oriented triangulations around a typical edge and is expected to be in the LQG universality class with $\gamma=\sqrt{4/3}$. We first explain the analogies between this model and last and first passage percolation. Then, we construct the Busemann function, which measures directed distance to infinity along a natural interface of the UIBOT. We show that, in the case of longest (resp. shortest) directed paths, this Busemann function converges in the scaling limit to a $2/3$-stable Lévy process (resp. a $4/3$-stable Lévy process). These results imply that in a typical subset of the UIBOT with $n$ edges, longest directed path lengths are of order $n^{3/4}$ and shortest directed path lengths are of order $n^{3/8}$. 

 We conclude the talk by explaining why these results fit into a program to construct the (longest and shortest) directed LQG metrics, two distinct two-parameter families of random fractal directed metrics which generalize the LQG metric and which could conceivably converge to the directed landscape upon taking an appropriate limit. 

Based on joint work with E. Gwynne.