  A $k$-differential on a Riemann surface is a section of the $k$-th power of
the canonical bundle. Loci of $k$-differentials with prescribed number and
multiplicities of zeros and poles form a natural stratification for the moduli
space of $k$-differentials. The classification of connected components of the
strata of $k$-differentials was known for holomorphic differentials,
meromorphic differentials and quadratic differentials with at worst simple
poles by Kontsevich--Zorich, Boissy and Lanneau, respectively. Built on their
work we develop new techniques to study connected components of the strata of
$k$-differentials for general $k$. As an application, we give a complete
classification of connected components of the strata of quadratic differentials
with arbitrary poles. Moreover, we distinguish certain components of the strata
of $k$-differentials by generalizing the hyperelliptic structure and spin
parity for higher $k$. We also describe an approach to determine explicitly
parities of $k$-differentials in genus zero and one, which inspires an amusing
conjecture in number theory. A key viewpoint we use is the notion of
multi-scale $k$-differentials introduced by
Bainbridge--Chen--Gendron--Grushevsky--M\"oller for $k = 1$ and extended by
Costantini--M\"oller--Zachhuber for all $k$.
