Column Base Connections (CBCs) are one of the most important components of Steel Moment Frames (SMFs) since these connections transfer the loads (e.g., gravity, wind, or earthquake loads) from the superstructure to the foundation. Because of it, CBCs have been extensively studied over the last decade through large-scale experimental programs and computational models that have led to the development of well-developed design guides such as the Design Guide 1 (DG1). Early analytical models were developed to predict the strength of CBCs, while subsequent studies estimated their rotational Stiffness, as well as deformation capacity. Modern studies on the topic have been developed to explore their behavior further with sophisticated finite element simulations. Moreover, their high deformation capacity with desirable hysteretic properties has called the researcher's attention to incorporate CBCs as part of the energy dissipation system.The broad research conducted on this area points to the importance of CBCs. Motivated by this issue, this paper summarizes the (state of the art) analytical and numerical models that characterize the behavior of CBCs. First, the different types of CBCs used in the industry are described. Next, the analytical models to predict their strength, rotational Stiffness, and hysteretic properties are summarized. Strategies for the development of finite element models, as well as results from simulations, are detailed. The influence of column base behavior on the global performance of SMFs (varying in height) is described. Finally, the author's essential results from ongoing research on ductile base plates are discussed, and lines for future investigations are recommended.