Large-scale reactor calculations with Monte Carlo (MC), including nonlinear feedback effects, have become a reality in the course of the last decade. In particular, implementations of coupled MC and thermalhydraulic (T-H) calculations have been separately developed by many different groups. Numerous MC codes have been coupled to a variety of T-H codes (system level, subchannel, and computational fluid dynamics). In this work we review the numerical methods that have been used to solve the coupled MC-T-H problem with a particular focus on the formulation of the nonlinear problem, convergence criteria, and relaxation schemes used to ensure stability of the iterative process. We use a simple pressurized water reactor pin cell problem to numerically investigate the stability of commonly used schemes and which problem parameters influence the stability-or lack thereof. We also examine the role that the running strategy used in the MC calculation plays in the convergence of the coupled calculation. Results indicate that the instability in fixed-point iterations is driven by the Doppler feedback effect and that underrelaxation can be used to restore stability. We also observed that a form of underrelaxation could be achieved by performing the coupled iterations without converging the MC fission source each iteration. By performing many iterations of few histories, we observed rapid convergence to the coupled MC-T-H solution in a relatively small number of batches. Numerical results also showed that the presence of instability in the fixed-point iteration is independent of the stochastic noise in the MC simulation.