The salient advance in electronics science has occurred due to the huge progress in the electronic devices fabrication technology in Nano and Micro scales in recent years. Memristor is a new electronic component which has been recently discovered and can be fabricated in sub-micron dimensions. A working prototype of memristor was announced by HPCompany, for the first time, in 2008. Since then, it has drawn many research centers attentions from all over the world. In this thesis, first memristor is introduced and its electrical characteristics and applications in different areas are described. Then the performances of different technologies in which memristor somehow are used are compared to each other. After introducing several main structures designed by using memristor, the proposed circuits and memories are studied and their advantages compared with similar works are discussed. The feasibility of using memristors to implement multilevel memory and logic is one of the most important advantages of these devices that are presented in this work. It is shown that it can improve some important memory parameters such as read speed, power consumption, and occupied area of the chip in an integrated circuit technology. Keywords: Memristor Circuits,Nonlinear Element, Multilevel Architecture, Memory.