A package stack structure includes a lower semiconductor chip on a lower package substrate having a plurality of lower via plug lands, a lower package having a lower molding compound surrounding a portion of a top surface of the lower package substrate and side surfaces of the lower semiconductor chip, an upper semiconductor chip on an upper package substrate having a plurality of upper via plug lands, an upper package having an upper molding compound covering the upper semiconductor chip, via plugs vertically penetrating the lower molding compound, the via plugs connecting the lower and upper via plug lands, respectively, and a fastening element and an air space between a top surface of the lower molding compound and a bottom surface of the upper package substrate.