In this thesis an approach to solving the sign problem in various quantum field theoretic models on the lattice, based on an exact reformulation -- the so called dualization -- is presented. The sign problem most prominently shows up in field theories at non-zero density and spoils numerical simulations with the conventional degrees of freedom. A dualization maps these degrees of freedom to new, dual variables, which allow for a formulation of the partition function of the model with real and non-negative weights only and thus, makes Monte Carlo simulations feasible. Four large classes of models with sign problems are discussed and their dual representations are derived. Numerical results are obtained on the one hand as a proof of concept and on the other hand to study previously inaccessible regions in their respective parameter spaces. The corresponding physical results are discussed in detail. Besides allowing for numerical studies, dual representations also give a clue about the physically important field configurations and allow in many cases for an intuitive interpretation of the field-particle correspondence. This interpretation is used to derive new methods for the study of scattering processes.