Abstract-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 applications. The first antenna has directional pattern and operates at 1760, 2550 and 3850 MHz (three-band antenna) with gain 2.1, −3.9 and 2.5 dBi, and it is dual polarized. The size of prototype patch antenna is 20 × 20 mm 2 which is reduced about 47% in comparison to conventional patch antenna at 2.5 GHz. The second antenna is designed by the use of interdigital capacitor and spiral inductor. Dimensions of antenna are 15.5×12 mm 2 , so the size is reduced about 69% in comparison to conventional microstrip patch antennas at 1.8 GHz. The second tri-band antenna operates at 1060 MHz, 1800 MHz and 2500 MHz in which two frequencies (1.8 and 2.5 GHz) are suitable for GMS and WLAN applications. Both structures have been designed and fabricated on FR4 low cost substrate with ε r = 4.4 and thickness of 1.6 mm. All simulations are done with CST and HFSS. Equivalent circuit and experimental results are also presented and compared.