Emerging computing applications (such as big-data and Internet-of-things) are extremely demanding in terms of storage, energy and computational efficiency, while today’s architectures and device technologies are facing major challenges making them incapable to meet these demands. Computation-in-Memory (CIM) architecture based on memristive devices is one of the alternative computing architectures being explored to address these limitations. Enabling such architectures relies on the development of efficient memristive circuits being able to perform logic and arithmetic operations within the non-volatile memory core. This paper addresses memristive circuit designs for CIM architectures. It gives a complete overview of all designs, both for logic as well as arithmetic operations, and presents the most popular designs in details. In addition, it analyzes and classifies them, shows how they result in different CIM flavours and how these architectures distinguish themselves from traditional ones. The paper also presents different potential applications that could significantly benefit from CIM architectures, based on their kernel that could be accelerated. @en