Mobile telephone systems have become hugely popular over the last decade. The development of mobile telephone system has been rapid and diversified in order to increase its marketability. GSM (Global System for Mobile) and PCS (Personal Communication System) are two of the most popular mobile telephone networks and their spectrum frequencies are located around 900MHZ and 1800MHz respectively. One of the recent trends is the combination of these two systems, which can effectively enhance the network coverage. Another new development is the design of an internal antenna, which includes the Patch and the planar inverted F antenna (PIFA). The key advantages of the Patch and PIFA include the prevention of radiation diverted towards users' heads and the reduction in the size of the antenna. The design of a dual band antenna incorporates these two recent developments. The purpose of this thesis is to propose different designs of compact dual-band antenna that can be employed in wireless mobile telephones. Two types of novel antenna designs are analyzed, including Dual Band RF Trap antenna and Dual Band MIFA RF Trap antenna. The volume of the two antennas is 25 applicable for both GSM and DCS1800 systems. The design of the two antennas is based on that of the PIFA antenna so the radiation diverted towards the user head can be controlled. Possessing these advantages, dual-band antennas are proved to have potentials for further development in the mobile telephone market.