In this review, we assess the use of Bayesian methods in model predictive control (MPC), focusing on neural-network-based modeling, control design, and uncertainty quantification. We systematically analyze individual studies and how they are implemented in practice. While Bayesian approaches are increasingly adopted to capture and propagate uncertainty in MPC, reported gains in performance and robustness remain fragmented, with inconsistent baselines and limited reliability analyses. We therefore argue for standardized benchmarks, ablation studies, and transparent reporting to rigorously determine the effectiveness of Bayesian techniques for MPC.