In this paper, periodic structures are investigated in antenna design for wireless applications. These antennas were compared with CRLH miniaturization method. Three different models of patch antenna with coaxial feed on EBG ground, metamaterial substrate or EBG/AMC structure have been presented here. Also two compact dual-band antennas have been designed and fabricated based on CRLH techniques for wireless and GSM (Global System for Mobile communication) applications. The first antenna has directional pattern and operates at 1760, 2550 and 3850MHz (three-band antenna) with gain 2.1, -3.9 and 2.5dBi and it is dual polarized.