  Multifractal behavior has been identified and mathematically established for
large classes of functions, stochastic processes and measures. Multifractality
has also been observed on many data coming from Geophysics, turbulence,
Physics, Biology, to name a few. Developing mathematical models whose scaling
and multifractal properties fit those measured on data is thus an important
issue. This raises several still unsolved theoretical questions about the
prescription of multifractality (i.e. how to build mathematical models with a
singularity spectrum known in advance), typical behavior in function spaces,
and existence of solutions to PDEs or SPDEs with possible multifractal
behavior. In this survey, we gather some of the latest results in this area.
Dedicated to Alain Arn{\'e}odo, pioneer in the development of wavelet tools for
data analysis.
