"Planar Ising model, uniform spanning trees, and bipartite dimers are classical examples of free fermionic models in two-dimensional statistical physics. Given a planar graph carrying such a model at or near its critical point – or, a sequence of such graphs with mesh size going to zero – one is interested to find a relevant complex structure that describes the behavior of correlation functions. Recently, it was understood that in many cases this description most naturally comes from discrete two-dimensional surfaces, so-called s- or t-surfaces, embedded into R^{2,1} (Ising model) or R^{2,2}, respectively. I plan to review basic constructions and results obtained on the statistical physics side as well as important open questions about these discrete surfaces."