Intracellular transport is one of the most confusing issues in the field of cell biology. Many different models and their combinations have been proposed to explain the experimental data on intracellular transport. Here, we analyse the data related to the mechanisms of endoplasmic reticulum-to-Golgi and intra-Golgi transport from the point of view of the main models of intracellular transport; namely: the vesicular model, the diffusion model, the compartment maturation–progression model, and the kiss-and-run model. This review initially describes our current understanding of Golgi function, while highlighting the recent progress that has been made. It then continues to discuss the outstanding questions and potential avenues for future research with regard to the models of these transport steps. To compare the power of these models, we have applied the method proposed by K. Popper; namely, the formulation of prohibitive observations according to, and the consecutive evaluation of, previous data, on the basis on the new models. The levels to which the different models can explain the experimental observations are different, and to date, the most powerful has been the kiss-and-run model, whereas the least powerful has been the diffusion model.