Computer arithmetic has been an active area of research since the advent of computers. The number system study diverged to fit the underlying computer architecture and applications. The acceleration of arithmetic circuits has been a challenging task due to the complexities involved in hardware designs. The advances in technologies, backed by innovations, led to the performance improvements in the sequential arithmetic circuits until the breakdown of Moore’s law. There was significant progress in the meantime in the domain of number representations to fit maximum information per bit. In the late 1990s and early 2000s, the inventions in energy and area efficient arithmetic circuits flourished with growing application requirements. The three formats, integer, fixed-point, and floating-point, became prominent based on application requirements, and several approximation techniques were exercised in the representations and across the representations. Post-2010, approximate computing rose to prominence as several applications could withstand “enough” precision and numerical accuracy of the arithmetic. While the approximate computing completely undermined the reproducibility aspects of the arithmetic, rendering high-performance gains, this chapter will discuss an overall overview of the current state and future directions for computer arithmetic and arithmetic architectures.