A fusene is a plane graph in which all bounded faces are hexagons, all vertices are of degree 2 or 3 while all vertices not in the boundary are of degree 3. A self-avoiding polygon in the honeycomb lattice is an example of a fusene, but a general fusene should be thought of as a lattice polygon that is allowed to self-overlap. One may also interpret a fusene as a flat hexagonal tiling of a topological disk and this way equipping it with a flat metric. In this talk I will discuss the problem of enumeration of fusenes, and other classes of self-overlapping lattice polygons, and highlight some properties of the random flat metric on the disk associated to uniform random self-overlapping lattice polygons of fixed large perimeter. This is a problem that arises naturally in the physics context of JT gravity, as pointed out in recent works by F. Ferrari.