Have you seen Manin’s book “Cubic forms: algebra, geometry, arithmetic”?

A form of degree greater than $2$ is a homogeneous polynomial of degree greater  $2$. Of course there are important examples of such polynomials (e.g., the determinant polynomial of degree $n$), but a major distinction between $n=2$ and $n>2$ is that in degree $2$ the orthogonal group of a quadratic form can be quite intricate, while in higher degree the linear transformations preserving a form $f$ form a finite group if it has degree $n$ at least $3$, nonzero discriminant, and we are very a field of characteristic $0$ or characteristic bigger than $n$ (theorem of Jordan classically). 

Over $\mathbf C$, the only linear transformations preserving $\sum_{i=1}^d x_i^d$ for $d>2$ is compositions of coordinate permutations and scaling of coordinates by $d$th roots of unity (a finite group of order $2^d d!$).

For $d>2$, forms of degree $d$ often can’t be diagonalized.