Abstract: The dimer model is a statistical mechanics model that studies random perfect matchings on graphs. It was introduced in the 1960s, when the partition functions and correlations were exactly computed by Kasteleyn, Temperley, and Fisher. Over the past few decades, significant progress has been made in the bipartite case. In this talk, I will characterize the Newton polygons associated with the characteristic polynomials arising from the dimer model of certain families of non-bipartite graphs. I will also introduce some new local moves on non-bipartite graphs that preserve the dimer partition function. Finally, I will talk about recent results regarding the enumeration of perfect matchings on some 3-regular graphs.