In this talk I will be speaking about the dimer model sampled on a general Riemann surface. In this setup, the dimer height function becomes additively multivalued with a random monodromy. Given a sequence of graphs approximating the conformal structure of the surface in a suitable way, the underlying sequence of height functions is expected to converge to the compactified free field on the surface. Recently, this problem was addressed by Berestycki, Laslier and Ray in the case when a Riemann surface is approximated by Temperley graphs. Using various probabilistic methods, they obtained the following universal result: given that the random walk associated with these graphs converges to the Brownian motion on the surface (in an appropriate sense), the limit of height functions exists, is conformally invariant and does not depend on a particular sequence of graphs. However, the identification of the limit with the compactified free field was missing in this result. In my recent work I am trying to fill this gap by studying the same problem from the perspective of discrete complex analysis. For this purpose, I consider graphs embedded into locally flat Riemann surfaces with conical singularities and satisfying certain local geometric conditions. In this setup I obtain an analytic description of the limit which allows to identify it with a suitable version of the compactified free field; I also prove the convergence in some non-Temperlian cases when the surface is generic. A core part of this approach is the regularity theory on t-embeddings recently developed by Chelkak, Laslier and Russkikh, as well as an analytic technique linking the problem with Quillen determinant of a family of Cauchy-Riemann operators developed by Dubédat.