To date, more than 5000 exoplanets and more than 2000 brown dwarfs have been confirmed, which shows rich diversities in many aspects. With the rapid growth of the planet family, both observational and theoretical research on exoplanet atmosphere is developing rapidly. This paper aims to review the progress of exoplanet atmospheric circulation research, and these atmospheric circulation characteristics are the basis for exoplanet and brown dwarf exploration in the future. First, this paper will review the main progress on exoplanet and brown dwarf detection in recent decades, and introduce the basic characteristics of the atmospheric circulation of hot Jupiters, terrestrial planets with atmosphere, and isolated and rapidly rotating brown dwarfs. Then, the simulation results of the atmospheric evolution and dynamic characteristics of exoplanets based on the atmospheric observations will be introduced, including the vertical temperature distribution of the atmosphere, the stability of the stratification, the radiation transfer, and the atmospheric composition, etc. Hot Jupiters are strongly irradiated and tidally locked. The general circulation models of hot Jupiters predict large day-night temperature differences and equatorial eastward jets. Warm Jupiters are generally not tidally locked, exhibiting a wide range of inclinations, orbital eccentricities and rotation rates, resulting in very different circulation patterns compared with hot Jupiters. Atmospheric circulation patterns of tidally locked terrestrial planets are similar to those of hot Jupiters, however, the existence of oceans and various chemical processes will largely change the climate of terrestrial planets. Besides, in the part of terrestrial planets, we will discuss the ultimate goal of exoplanet research—searching and characterizing habitable planets. As a transition state between planets and stars, brown dwarfs are rapidly rotating without external irradiation. Their interiors are fully convective, and their atmosphere exhibits clouds and temperature inhomogeneities. Finally, an outlook will be given to the key issues and challenges of exoplanet and brown dwarf atmosphere.