Strongly-interacting theories lie at the heart of elementary particle physics. Their distinct behaviour shapes our world sui generis. We are interested in lattice simulations of supersymmetric models, but every discretization of space-time inevitably breaks supersymmetry and allows renormalization of relevant susy-breaking operators. To understand the role of such operators, we study renormalization group trajectories of the nonlinear O(N) Sigma model (NLSM). Similar to quantum gravity, it is believed to adhere to the asymptotic safety scenario. By combining the demon method with blockspin transformations, we compute the global flow diagram. In two dimensions, we reproduce asymptotic freedom and in three dimensions, asymptotic safety is demonstrated. Essential for these results is the application of a novel optimization scheme to treat truncation errors. We proceed with a lattice simulation of the supersymmetric nonlinear O(3) Sigma model. Using an original discretization that requires to fine tune only a single operator, we argue that the continuum limit successfully leads to the correct continuum physics. Unfortunately, for large lattices, a sign problem challenges the applicability of Monte Carlo methods. Consequently, the last chapter of this thesis is spent on an assessment of the fermion-bag method. We find that sign fluctuations are thereby significantly reduced for the susy NLSM. The proposed discretization finally promises a direct confirmation of supersymmetry restoration in the continuum limit. For a complementary analysis, we study the one-flavor Gross-Neveu model which has a complex phase problem. However, phase fluctuations for Wilson fermions are very small and no conclusion can be drawn regarding the potency of the fermion-bag approach for this model.