M-dwarf systems provide several compelling motivations for study. M-dwarf stars are the most numerous, the longest-lived, and the most amenable targets for detecting and characterizing extrasolar planets. Furthermore, recent discoveries of Proxima Centauri b and seven terrestrial sized planets orbiting TRAPPIST-1 provide tangible evidence for habitable zone planets around M-dwarf stars. While observational detail remains sparse, we must fill in the gaps using theoretical models. However, the nuances of these systems pose unique challenges for atmospheric modeling. M-dwarf stars are small, dim, and red. Each of these factors has important consequences for the climate and habitability of terrestrial planets. Habitable zone planets in M-dwarf systems must orbit quite close to their host star in order to receive sufficient stellar radiation, and thus they are likely tidally locked. This imposes a profound control on planetary climate, as the orbital period, planetary rotation rate, incident stellar flux, and spectral energy distribution become inextricably linked. Atmospheric dynamics, convection, clouds, and shortwave absorption by water vapor all are affected. Traditional 1D radiative-convective climate models fundamentally cannot simulate tidally locked and slow rotating planets that are expected in the habitable zones of M-dwarf stars. One must use 3D general circulation and climate system models in order to adequately simulate these worlds. In this seminar, I will discuss recent advances in climate modeling of terrestrial extrasolar planets orbiting M-dwarf stars.