In this chapter, the fractal functions constructed as attractors of iteration function systems, referred as fractal interpolation functions, are discussed in detail. In Chap. 1, some examples of fractal functions such as Weierstrass function and Takagi function are discussed, they are called fractal functions due to their non-differentiable nature and non-integer fractal dimension. However, they are not constructed using the IFS theory. In 1986, Barnsley [1] has developed the generalized interpolation functions with fractal characteristics and has beautifully narrated their greater efficiency in comparison with the classical (Euclidean) interpolation functions. In general, interpolation is a technique to estimate the missing value of a function \(f : [a, b] \rightarrow \mathbb {R}\) at any given point \(x \in [a, b]\) such that whose graph passes through all the given finite number of points \(a = x_1 < x_2 < \cdots < x_N = b\) . Suppose an experiment gives the data points \(\{(0,0.1),(0.25,1.1),(0.5,-0.4),(0.75,0.8),(1,0.3)\}\) to be investigated. Then, the traditional method for analyzing the data begins with the graphical representation. For instance, a linear interpolation and polynomial fit with degree 4 are used for the approximation and the process is illustrated in Fig. 2.1a and b.