This Thesis discusses how non-chiral topological order in two-dimensions can be accessed within a tensor network framework. The approach is two-fold. On the one hand, we extensively discuss how tensor networks can be used as a tool to characterize topological order and how this approach blends in with other mathematical frameworks, such as string-net models, state sum constructions and the abstract algebraic theory of anyons formulated using category theory. On the other hand we use the tensor network description as a conceptual tool in the design of synthetic topological quantum matter built from mesoscopic devices, concretely networks of tunnel coupled Majorana Cooper boxes.
