Transport-dominated systems are characterized by the propagation of waves and occur in many applications such as aerodynamics and chemical engineering. To predict the dynamics of such systems, mathematical models should ideally be fast to evaluate and at the same time sufficiently accurate. One possibility for deriving such models is to start with a complex and accurate full-order model (FOM) and use model order reduction (MOR) techniques to obtain a corresponding reduced-order model (ROM). Classical MOR methods are based on approximating the FOM state by a linear combination of ansatz functions or modes, but such approaches are often inadequate in the context of transport-dominated systems. This is one of the reasons why there has been an increasing research effort in the past years to develop MOR techniques which are based on nonlinear approximation ansatzes.As the field of nonlinear MOR is relatively new, there are still many open research questions to be addressed. These include for instance suitable choices for the approximation ansatz as well as appropriate ways for the construction of corresponding ROMs. Furthermore, nonlinear MOR approaches typically lead to ROMs whose evaluation scales with the dimension of the FOM and thus may be too expensive. In fact, similar issues may also occur in the context of linear MOR approaches and, therefore, one uses so-called hyperreduction techniques to obtain fast ROMs. However, classical hyperreduction methods suffer from similar difficulties as classical MOR schemes when being applied to transport-dominated systems. Another challenge is to develop nonlinear MOR techniques which preserve important system properties such as stability.In this thesis, we present a new nonlinear model reduction framework which is based on approximating the state of the FOM by a linear combination of transformed modes. The transformations may be, e.g., achieved by shift operators and are parametrized by so-called paths or shift amounts, which constitute a part of the ROM state. The resulting class of ansatzes is well-suited for obtaining low-dimensional and accurate approximations of transport-dominated systems. For the determination of the modes, we present an optimization approach based on given snapshot data of the FOM state. Furthermore, the construction of the ROM is carried out via a residual minimization approach and we also suggest a new hyperreduction framework to ensure that the ROM can be efficiently evaluated. In addition, we demonstrate how to preserve stability via an energy-based formulation using the framework of so-called port-Hamiltonian systems. Finally, we illustrate the new methodology by means of numerical experiments for some transport-dominated test cases.