This thesis is focused on the modelling of superconducting materials, and in particular, high-temperature superconducting materials. This work is divided in two parts: first, the dynamic magnetic field mapping obtained by measurements is used in order to reconstruct the dynamic current distribution inside a tape assuming a model describing its global behavior. In the second part, superconductors in over-critical regime have been both electrically and thermally modelized with a finite element method. Calculations have been applied on a current limiter and compared to a constructed and measured device at the University of Geneva. The first chapter is an introduction to high-temperature superconducting materials, where the main superconducting physical properties used are presented. Next, we introduce the different existing models which can be used for describing the behavior of a superconducting tape. With the help of the Bean model, a novel method to evaluate the current distribution in a mono-filamentary superconducting tape from surface magnetic field measurements is proposed. The obtained results are compared with more precise, but more complex, methods like finite element modeling, which will be used for the second part of this work. The last part is focused on the modelization of superconducting material in overcritical regime. To achieve this goal, we need to introduce a different expression than the usual Ec(J/Jc)n power-law for describing the electric behavior of the material in a much wider current range covering over-critical excursion. The original proposition we made is able to fit the measurements made on YBCO tapes, allowing to describe the transition from the superconducting to the normal state. Then, to taking into account the thermal phenomena, the temperature dependence of the electrical parameters has been introduced in order to solve a coupled electromagnetic and thermal problem: solving a time step of the electromagnetic part leads to the knowledge of the local losses. These losses are injected inside the thermal part. The resulting temperature computation is used to modify the electrical parameters for the next time step. Finally, this method has been applied in order to simulate a superconducting current limiter. The global behavior of the device can be reproduced by the implemented numerical model, which also allows the study of the local variables, like current density distribution or temperature profile. The obtained results can be used to optimize a device according to specific criteria. In particular, we have proposed geometric modifications for avoiding possible local thermal runaway, which can lead to the destruction of the device. This proposition has beeing inspired the new design, which is now tested.