The chapter “An Introduction to Multifractals and Scale Symmetry Groups” is designed to provide a broad and in-depth overview on the fundamentals, the current developments and future directions. It aims to make mutlifractals more useful and, in fact, more indispensable for geophysics, while too restricted formalisms have prevented their application to the full extent. This chapter therefore replaces the scale symmetries in the general framework of symmetry groups, as well as of the pullback and push forward transforms of fields and measures. This establishes a generalized scale invariance framework, having a much wider applicability than the too restricted framework of the classical self-similarity. The somewhat classical case of scalar-valued multifractals is reviewed. The two main multifractal formalisms, based respectively on dimensions and co-dimensions are shown to correspond respectively to deterministic and stochastic processes. Stochastic extremes are analyzed in terms of multifractal phase transitions and self-organised criticality. Statistical universality is discussed and shown to drastically reduce the number of relevant multifractal parameters to a few physically meaningful exponents. Finally, insights are given on a recent extension to vector-valued multifractals, which have been particularly needed in geophysics and constitute a prominent research direction.