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Elliptic surfaces play a prominent role in the classification of algebraic surfaces and fit well within the general theory. In fact, they admit relatively simple foundations which in particular do not require too many prerequisites. Yet elliptic surfaces are endowed with many different applications for several areas of research, such as lattice theory, singularities, group theory, and modular curves. Our aim is to present some recent advances in studying the structure of parameter (or moduli) spaces of elliptic surfaces. We introduce a combinatorial description which involves special graphs and triangulations embedded into the sphere in the spirit of the famous Belyi theorem. This construction allows a relatively transparent description of a rather complicated set of global monodromy groups of elliptic surfaces. In this way the general results of Kodaira theory are transformed to a concrete computational tool.