Chaotic systems have shown advantages in a wide range of applications from noise signal generators up to secure communications and neuromorphic systems. Such applications include the use of chaotic maps and fractional-order chaotic systems that can generate self-excited and hidden attractors, and as described in Chap. 1 , they can be implemented by using integrated circuit technology, commercially available electronic devices, microcontrollers, raspberry-Pi, and embedded hardware such as field-programmable analog arrays (FPAAs) and field-programmable gate arrays (FPGAs). In the implementations, numerical methods play an important role to guarantee chaotic behavior in a long time and to provide high throughput and maximize the operation frequency that can be achieved by using low hardware resources. This chapter shows the models of the chaotic systems that are case study in the following chapters and some numerical methods to solve integer- and fractional-order chaotic systems. The models consists of ordinary differential equations (ODEs) consisting of three (3D), four (4D), and five (5D) state variables. In particular, this book considers chaotic maps, the well-known Lorenz system, a multistable 3D memristive that is simulated in its integer- and fractional-order versions, and two hyperchaotic systems (4D and 5D). They are implemented on FPGA to design pseudo-random number generators (PRNGs), in Chaps. 5 and 6 .