Interpolation theory focuses on the existence of continuous functions which reconstruct the prescribed data set thereby helping to approximate the real-valued continuous function. In the historical development of the approximation theory, beyond polynomials, splines and trigonometric functions are likewise applied in approximation methods which are directed toward the smooth approximants. Then again, numerous practical and natural phenomenal signals reveal the non-smoothness in their traces and henceforth request non-smooth functions for significant reconstruction. The notion of a fractal function is explored based on the theory of iterated function system which was made by envisaging the universe as a fractal. Consequently, fractal functions are used for both smooth and non-smooth approximation by including various classical approximation methods, although there are differences between the fractal interpolation function and the traditional interpolation function (see for more details, [42–59]). Fractal interpolants are constructed by the theory of iterated function system which offers a self-referential functional equation for the interpolant and implies a self-similarity in magnification. Additionally, the choice of vertical scaling factors provides a flexible and optimal interpolant which also generates a specific traditional interpolant [47]. As a consequence of its fruitful success in non-smooth approximation, the theory of fractal functions has been extensively investigated, and there are developments in the pipeline beyond its mathematical framework. This chapter presents the construction of fractal interpolation functions and their evolution based on the vertical scaling factors and different iterated function systems. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.