  B. Mandelbrot gave a new birth to the notions of scale invariance,
selfsimilarity and non-integer dimensions, gathering them as the founding
corner-stones used to build up fractal geometry. The first purpose of the
present contribution is to review and relate together these key notions,
explore their interplay and show that they are different facets of a same
intuition. Second, it will explain how these notions lead to the derivation of
the mathematical tools underlying multifractal analysis. Third, it will
reformulate these theoretical tools into a wavelet framework, hence enabling
their better theoretical understanding as well as their efficient practical
implementation. B. Mandelbrot used his concept of fractal geometry to analyze
real-world applications of very different natures. As a tribute to his work,
applications of various origins, and where multifractal analysis proved
fruitful, are revisited to illustrate the theoretical developments proposed
here.
