According to one embodiment, an insulated structure includes a frame comprising a plurality of wall studs coupled together and a plurality of foam boards attached to the frame to form a continuous insulative wall. A plurality of fasteners attaches the foam boards to the frame. Each fastener includes an elongate shaft and a cap. The elongate shaft is configured to penetrate through a foam board and into a wall stud to couple the components together. The cap is configured to be positioned atop a foam board to distribute a load relatively evenly to the foam board. A sealing tape is applied across seams between adjacent foam boards and over the fasteners' caps to seal the wall. A sealing caulk is applied to secondary fasteners and penetrations to seal the wall. In some embodiments, the structure has a fastener density of about 1 fastener per 243 in2 of foam board.