It is not yet entirely clear whether Mars began as a warm and wet planet that evolved towards the present-day cold and dry body or if it always was cold and dry with just some sporadic episodes of liquid water on its surface. An important clue into this question can be gained by studying the earliest evolution of the Martian atmosphere and whether it was dense and stable to maintain a warm and wet climate or tenuous and susceptible to strong atmospheric escape. In this review we therefore discuss relevant aspects for the evolution and stability of a potential early Martian atmosphere. This contains the EUV flux evolution of the young Sun, the formation timescale and volatile inventory of the planet including volcanic degassing, impact delivery and removal, the loss of the catastrophically outgassed steam atmosphere, atmosphere-surface interactions, as well as thermal and non-thermal escape processes affecting a potential secondary atmosphere at early Mars. While early non-thermal escape at Mars before 4 billion years ago is poorly understood, in particular in view of its ancient intrinsic magnetic field, research on thermal escape processes and the stability of a CO2-dominated atmosphere around Mars against high EUV fluxes indicate that volatile delivery and volcanic degassing cannot counterbalance the strong thermal escape. Therefore, a catastrophically