This paper deals with a brief historical introduction to fractals, fractal dimension and fractal geometry. Many fractals including the Cantor fractal, the Koch fractal, the Minkowski fractal, the Mandelbrot and Given fractal are described to illustrate self-similar geometrical figures. This is followed by the discovery of dynamical systems and iterative mappings with the remarkable examples of the Julia fractals and the Mandelbrot fractals. Fractals, chaos, bifurcations and Hausdorff dimension seem to be essential elements of fractal geometry. Some applications of fractals to fracture mechanics and turbulence are briefly discussed. Included are Mumford's new computer aided explorations to visualize Kleinian groups of maps of the complex plane of fractional linear transformations. This leads to a family of extremely beautiful symmetrical fractal structures including Indra's net, Indra's necklace and Indra's pearls which describe Klein's mathematical vision. It is shown that modern computer technology and computer graphical analysis have the remarkable ability and power to generate totally new and unexpected complex fractal structures that are surprisingly very beautiful.