This work tried to highlight the potential of memristor in binary and multilevel digital circuits. This book focuses on the ternary and redundant circuits where the concept could be generalized for any multilevel circuits. Ternary half adder circuit was the first circuit to be investigated to address the concept of multilevel circuit based on the memristor. A complete case study for redundant half adder, full adder and N-bit adder circuit based on the memristor analyzed to build an adder that has speed independent on the operand showing the potential of this element in arithmetic circuit. New approach to build digital circuit using the memristor was also introduced, by replacing the complete pull down network with one memristor to work as calculating/saving element to decreasing the number of transistors. An example to implement redundant multiplier circuit based on this approach was then introduced. All works were validated using Orcad.