We present stellar evolution calculations for the lowest mass stars, i.e., those stars with masses in the range 0.08 M⊙ <= M* <= 0.25 M⊙. Our particular emphasis is on the post-main-sequence evolution of these objects. We establish a hydrogen-burning timescale of τH ~ 1.0 × 1013 years for the minimum-mass main-sequence star. This timescale determines the duration over which the light of our Galaxy is dominated by a conventional stellar contribution. We find that for masses M* < 0.25 M⊙, stars remain fully convective for a significant fraction of the duration of their evolution. The maintenance of full convection precludes the development of large composition gradients and allows the entire star to build up a large helium mass fraction. We find that stars with masses M < 0.20 M⊙ will never evolve through a red giant stage. After becoming gradually brighter and bluer for trillions of years, these late M dwarfs of today will develop radiative-conductive cores and mild nuclear shell sources; these stars then end their lives as helium white dwarfs. Our work has significant bearing on the general question of why stars become red giants. The fact that the lowest mass stars grow neither red nor giant as they evolve provides an important insight into this problem. Through both analytical and numerical arguments, we have determined that the development of low-mass red giants requires a combination of (1) increasing core luminosity, (2) the existence of molecular weight gradients between the core and the envelope, and (3) the presence of an atmospheric opacity which is an increasing function of temperature. Finally, we discuss the implications of our results with regards to the long-term fate and evolution of the Galaxy.