General purpose.Most books which deal with fractional derivative refer to the Riemann-Liouvile definition in terms of integral: one first defines integral and then one defines derivative. On the contrary, this book provides a systematic self-contained presentation of fractional calculus, via fractional difference, and expands a fractional differential calculus which is quite parallel to the Leibniz calculus (therefore the expression of fractional differential calculus) and which is also quite physically meaningful. Whilst the standard fractional calculus applies to differentiable functions only, the present calculus holds for both differentiable functions and non-differentiable functions. Summary of content. Theory and application of this fractional differential calculus Proposals for some new approaches to analytical mechanics, differential geometry in fractal space-time, fractional white noise calculus, and information theory. Readership. Any scientist who is interested in fractals and in the applications of fractional calculus to natural science, either for the appications or for the foundations of physics