  There are various different notions measuring extremality of hypergraphs. In
this survey we compare the recently introduced notion of the codegree squared
extremal function with the Tur\'an function, the minimum codegree threshold and
the uniform Tur\'an density.
  The codegree squared sum $\textrm{co}_2(G)$ of a $3$-uniform hypergraph $G$
is defined to be the sum of codegrees squared $d(x,y)^2$ over all pairs of
vertices $x,y$. In other words, this is the square of the $\ell_2$-norm of the
codegree vector. We are interested in how large $\textrm{co}_2(G)$ can be if we
require $G$ to be $H$-free for some $3$-uniform hypergraph $H$. This maximum
value of $\textrm{co}_2(G)$ over all $H$-free $n$-vertex $3$-uniform
hypergraphs $G$ is called the codegree squared extremal function, which we
denote by $\textrm{exco}_2(n,H)$. We systemically study the extremal codegree
squared sum of various $3$-uniform hypergraphs using various proof techniques.
Some of our proofs rely on the flag algebra method while others use more
classical tools such as the stability method. In particular, we
(asymptotically) determine the codegree squared extremal numbers of matchings,
stars, paths, cycles, and $F_5$, the $5$-vertex hypergraph with edge set
$\{123,124,345\}$.
  Additionally, our paper has a survey format, as we state several conjectures
and give an overview of Tur\'an densities, minimum codegree thresholds and
codegree squared extremal numbers of popular hypergraphs. We intend to update
the arXiv version of this paper regularly.
