This comprehensive, accessible and very popular textbook presents fractal geometry at a level accessible to a final year undergraduate mathematician or physicist whilst also providing a useful primer or reference for the research mathematician or scientist. The author has brought this new edition up to date with the needs of todays reader by including an introductory multifractal theory, random fractals and modern applications in finance and science. New research developments in the area are highlighted, such as porosity, and other much more sophisticated topics such as fractal aspects of conformal invariance, complex dimensions, and non-commutative fractal geometry. Whilst there is an emphasis of dimension in its various forms, other notions of fractality are also prominent.