Lithium ion batteries have shown great promise in elearical energy storage with enhanced energy density, power capacity, charge-discharge rates, and cycling lifetimes. However common fluid electrolytes consisting of lithium salts dissolved in solvents are toxic, corrosive, or flammable. Solid electrolytes with superionic conductivity can avoid those shortcomings and work with a metallic lithium anode, thereby allowing much higher energy densities. Here we present a novel class of solid electrolytes with three-dimensional conducting pathways based on lithium-rich anti-perovslcites (LiRAP) with ionic conductivity of sigma > 10(-3) S/cm at room temperature and activation energy of 0.2-0.3 eV. As temperature approaches the melting point, the ionic conductivity of the anti-perovskites increases to advanced superionic conductivity of sigma > 10(-2) S/cm and beyond. The new crystalline materials can be readily manipulated via chemical, electronic, and structural means to boost ionic transport and serve as high-performance solid electrolytes for superionic Li+ conduction in electrochemistry applications.