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The book is devoted to the scaling properties of financial time series. In particular, the book deals carefully with the empirical determination of the Hurst exponent. The main statistical features of the financial indexes are presented, along with a brief overview of the main concepts in probability theory and fractal geometry. Then the role of extreme events and correlations in affecting the behaviour of the Hurst exponent is explained through the analysis of exactly solvable self-similar random walks. Finally the reliability of the multiscaling observed in finance is investigated both from a theoretical and an empirical viewpoint. Since the main result holds under quite general assumptions, the conclusions can be generalized to time series coming from other fields of the complex system physics, like hydrology and geophysics. The book, avoiding excessive formalism, is intended for a wide range of readers.