Among the contenders in the new generation energy storage arena, all-solid-state batteries (ASSBs) have emerged as particularly promising, owing to their potential to exhibit high safety, high energy density and long cycle life. The relatively low conductivity of most solid electrolytes and the often sluggish charge transfer kinetics at the interface between solid electrolyte and electrode layers are considered to be amongst the major challenges facing ASSBs. This review presents an overview of the state of the art in solid lithium and sodium ion conductors, with an emphasis on inorganic materials. The correlations between the composition, structure and conductivity of these solid electrolytes are illustrated and strategies to boost ion conductivity are proposed. In particular, the high grain boundary resistance of solid oxide electrolytes is identified as a challenge. Critical issues of solid electrolytes beyond ion conductivity are also discussed with respect to their potential problems for practical applications. The chemical and electrochemical stabilities of solid electrolytes are discussed, as are chemo-mechanical effects which have been overlooked to some extent. Furthermore, strategies to improve the practical performance of ASSBs, including optimizing the interface between solid electrolytes and electrode materials to improve stability and lower charge transfer resistance are also suggested. This critical review presents the state of the art research progress, proposes strategies to improve the conductivity of solid electrolytes, discusses the chemical and electrochemical stabilities, and uncovers future perspectives for solid state batteries.