In the past five years, there has been significant research interest in the intrinsic magnetic topological insulator family compounds MnBi2+2nTe4+3n (where n = 0, 1, 2 …). In particular, exfoliated thin films of MnBi2Te4 have led to numerous experimental breakthroughs, such as the quantum anomalous Hall effect, axion insulator phase, and high-Chern number quantum Hall effect without Landau Levels. However, despite extensive efforts, the energy gap of the topological surface states due to exchange magnetic coupling, which is a key feature of the system's characteristic band structure, remains experimentally elusive. The electronic structure measured by angle-resolved photoemission (ARPES) shows significant deviation from ab-initio prediction and scanning tunneling spectroscopy measurements, making it challenging to understand the transport results based on the electronic structure. This manuscript reviews the measurements of the band structure of MnBi2+2nTe4+3n magnetic topological insulators using laser-based ARPES, focusing on the evolution of their electronic structures with temperature, surface and bulk doping, and film thickness. The aim of the review is to construct a unified picture of the electronic structure of MnBi2+2nTe4+3n compounds and explore possible control of their topological properties.