  In the first part of this contribution, we review the development of the
theory of scale relativity and its geometric framework constructed in terms of
a fractal and nondifferentiable continuous space-time. This theory leads (i) to
a generalization of possible physically relevant fractal laws, written as
partial differential equation acting in the space of scales, and (ii) to a new
geometric foundation of quantum mechanics and gauge field theories and their
possible generalisations. In the second part, we discuss some examples of
application of the theory to various sciences, in particular in cases when the
theoretical predictions have been validated by new or updated observational and
experimental data. This includes predictions in physics and cosmology (value of
the QCD coupling and of the cosmological constant), to astrophysics and
gravitational structure formation (distances of extrasolar planets to their
stars, of Kuiper belt objects, value of solar and solar-like star cycles), to
sciences of life (log-periodic law for species punctuated evolution, human
development and society evolution), to Earth sciences (log-periodic
deceleration of the rate of California earthquakes and of Sichuan earthquake
replicas, critical law for the arctic sea ice extent) and tentative
applications to system biology.
