Dually weighted graphs (DWG) are planar Feynman graphs bearing two sets of couplings: one set of usual couplings $t_n$ attached to the vertices of valence $n$, and another set of dual couplings $t_n^*$ attached to the faces (dual vertices) of valence $n$. Such couplings allow a deep control on possible shapes of planar graphs. For example, if one turns on only the couplings $t_4$ and $t_4^*$ the graph takes a ''fishnet form'' of a regular square lattice. The problem of counting of such graphs can be formulated as a modified hermitian one matrix model with an extra constant matrix. The partition function can be then represented in terms of the ''character expansion'' over Young tableaux, solvable by the saddle point approximation. I will review old results on DWG from my papers with M.Staudacher and Th.Wynter, including the techniques of computing Schur characters of a large Young tableau and deriving the elliptic algebraic curve for counting of planar quadrangulations. Then I will present new results from our ongoing work with F.Levkovich-Maslyuk where we count the disc quadrangulations with large, macroscopic area and boundary. This allows to extract interesting continuous limit of fluctuating 2d geometry, interpolating between the ''almost'' flat disc with a few dynamical conical defects and the disc partition function for pure 2d quantum gravity, generalizing old results for the spherical topology. ---- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.