In many application fields, non-destructive testing and evaluation (NDT&E) is useful to determine material properties and detect faults of objects before or during operation. In the past decades, a wide range of experimental NDT&E techniques has been developed and successfully tested for many problem classes. In the past few years, the collection of experimental NDT&E techniques has been extended with hybrid methods, i.e., methods that combine experiments and numerical simulation. The main drawback of all existing NDT&E methods, is that they need a-priori knowledge and expertise of the operator. In this work, we aim to develop automated procedures for the characterisation of material parameters based on the simulation of wave propagation problems, thereby extending the application range of NDT&E, especially to problems with little a-priori knowledge. We propose the use of gradient based optimisation methods to develop automated procedures for the characterisation of material parameters. To achieve this, we first select efficient numerical procedures for the solution of wave propagation problems. For the spatial discretisation, we consider the finite integration technique and the finite element method with both continuous and discontinuous elements. We study the performance of several time integration methods in combination with these spatial discretisation methods, from which we learn that third order discontinuous elements in combination with second order time stepping methods are performing best. We compute gradient information of the time-dependent wave propagation problem and use this in combination with the forward solver to efficiently solve the inverse problems that occur in NDT&E. For the efficient computation of gradients, adjoint methods are used. The proposed gradient based optimisation method is compared to gradient free methods for determining the wave speeds in a homogeneous isotropic medium, from which we conclude that gradient based methods are the only feasible option to tackle more complicated inverse problems. The challenges involved in solving optimisation problems with spatially dependent control parameters are described and successfully demonstrated for determining a spatially dependent wave speed distribution. We show that the use of proper function spaces and associated Riesz maps is a key ingredient to obtain correct results. We also use gradient information to automatically calibrate absorbing layers to mimic open boundaries. The automatic calibration is performed for perfectly matched layers and for the simpler and computationally faster approach of using consecutive absorbing layers. We show that after calibration, both approaches have a comparable performance. Throughout this work, we use a high level of genericity to describe and simulate wave propagation problems so that all presented methods are applicable to acoustic, electromagnetic and elastic wave propagation problems in one two and three dimensions. This work demonstrates that numerically solving inverse NDT&E problems with many, e.g. spatially dependent, parameters is possible.