A key topic in systems biology is to understand the intricate relations between molecular network structures, dynamic properties and biological function. In this context, gene regulatory networks (GRNs) describing the regulatory interactions between genes and their products are of crucial importance. The general goal of this thesis is to explore the relationship between the structure and dynamics of GRNs. This is done in a discrete modelling framework using the Thomas formalism. A GRN is represented by a discrete model which includes an interaction graph (IG) and a logical parameter function that characterise the regulatory interactions. The dynamics of a GRN is modelled by an asynchronous state transition graph (ASTG), where the states of the system can only be changed by asynchronous and unitary updates. In 2011, T. Lorenz proposed two reverse engineering algorithms for inferring from a given ASTG models satisfying specific properties. In the first part of the thesis, the focus is on the explanation, implementation, and generalisation of the Lorenz algorithms. In order to handle general inputs, three necessary and sufficient conditions are presented to characterise ASTGs among all graphs on a given state space. Furthermore, a fourth condition is derived which is necessary and sufficient for an ASTG to admit a realistic model. These four ASTG conditions provide the basis for a generalisation of Lorenz algorithms and several applications. Multistationarity and homeostasis are two important dynamical properties of high biological relevance, which can be represented by attractors in the ASTG. In the second part of the thesis, two discrete modelling workflows are developed for exploring all those GRNs that are able to realise a given functionality. The forward modelling workflow includes enumerating all possible models and searching for those models whose ASTG exhibits the desired properties. The reverse engineering workflow starts from enumerating all graphs on the state space satisfying the dynamic properties and then infers all models of these graphs using the generalised Lorenz algorithms. To analyse the resulting functional IGs, a logical analysis method is developed, which encodes IGs by Boolean expressions, and then uses Boolean function minimisation to obtain a compact representation. The same logical analysis method can also be applied to the logical parameters. In the last part of the thesis, the discrete modelling workflows are applied to explore the space of GRNs realising some typical dynamic behaviours of biological interest. Three case studies are presented. The first one concerns homeostasis in a simplified MAPK cascade, the second one multistationarity in cell differentiation, and the third one single-stripe forming in the embryogenesis of the fruit fly Drosophila melanogaster.