Appears in collection : Jean Morlet Chair - Geometric and Probabilistic Aspects in Quantum Field Theory / Chaire Jean Morlet Conférence - Aspects géométriques et probabilistes de la théorie quantique des champs

I review the construction of JT quantum gravity on finite-size geometries, both from the combinatorial and from the probabilistic (path integral/CFT) points of view. Important open questions will be emphasized. Remarks on the construction of the path integral measure for JT gravity on infinite-size geometries (the Schwarzian theory) and for topological gravity will also be made.