  We develop the concept of fractal metamaterials which consist of arrays of
nano and micron sized rings containing Josephson junctions which play the role
of "atoms" in such artificial materials. We show that if some of the junctions
have $\pi$-shifts in the Josephson phases that the "atoms" become magnetic and
their arrays can have tuned positive or negative permeabilty. Each individual
"$\pi$-ring" - the Josephson ring with one $\pi$-junction - can be in one of
two energetically degenerate magnetic states in which the supercurrent flows in
the clockwise or counter-clockwise direction. This results in magnetic moments
that point downwards or upwards, respectively. The value of the total
magnetization of such a metamaterial may display fractal features. We describe
the magnetic properties of such superconducting metamaterials, including the
magnetic field distribution in them (i.e. in the network that is made up of
these rings). We also describe the way that the magnetic flux penetrates into
the Josephson network and how it is strongly dependent on the geometry of the
system.
