  We informally review the construction of spacetime geometries with
multifractal and, more generally, multiscale properties. Based on fractional
calculus, these continuous spacetimes have their dimension changing with the
scale; they display discrete symmetries in the ultraviolet and ordinary
Poincar\'e symmetries in the infrared. Under certain reasonable assumptions,
field theories (including gravity) on multifractional geometries are generally
argued to be perturbatively renormalizable. We also sketch the relation with
other field theories of quantum gravity based on the renormalization group.
