The history of describing natural objects by geometry is as old as the advent of science itself. Traditionally lines, squares, rectangles, circles, spheres etc. have been the basis of our intuitive understanding of the geometry. However, nature is not restricted to such Euclidean objects only which are characterized typically by integer dimensions. Most of the natural objects around us are so complex in shape that conventional Euclidean or integer dimension is not sufficient to describe them. Unfortunately, the early stage of our education teaches that object which have only integer dimension. Why are we jumping to integer dimensions only? Are there existing object with non-integer dimension. The present chapter is motivated by the desire from this thirst. This chapter deals the mathematical foundation for constructing fractals which has non-integer dimensions, the idea of fractal geometry appears to be indispensable for characterizing complex objects at least quantitatively. Further, fractals are generated as a unique invariant point by using Banach fixed point theorem in the complete metric space. This chapters ends with the examples which are resemblance with the natural objects.