Fractal curves and fractal surfaces have attracted considerable attention due to their applications in areas such as Metallurgy, Earth Sciences, Surface Physics, Chemistry and Medical Sciences. Fractal interpolation provides a new insight to the approximation and modelling of experimental data. Fractal interpolation curves and surfaces are usually considered as attractors of some iterated function systems associated with a given set of data. There are fixed point theorems that can be used for the existence of the attractor of an appropriately chosen iterated function system and to the generation of fractal interpolation functions. In this book, in order to give nonlinear fractal interpolation curves and surfaces with extensive practical applications, we use Rakotch contractions and the Matkowski fixed point theorem. In particular, we extend our work and offer a more comprehensive overview including some of the latest research made by the authors.