Similarly, as for smooth minimal surfaces, a discrete minimal surface may locally be constructed from a given discrete conformal map. However, there exists meanwhile a variety of definitions for discrete conformality and thus for the construction of discrete conformal maps and corresponding discrete minimal surfaces. In my talk, I will focus on two construction principles for discrete minimal surfaces, namely from boundary values of the smooth conformal map (Dirichlet problem) and from a real-analytic framed curve on the smooth minimal surface (Björling problem). For both cases, I explain the choice of a suitable definition for discrete conformality and the corresponding discrete construction principles. Furthermore, the main ideas of the proofs for the (local) approximation properties of the discrete and smooth conformal maps and minimal surfaces are presented. The approximation results for the Björling problem are joint work with Daniel Matthes.