  We establish a structure theorem for degree three codimension one foliations
on projective spaces of dimension $n\ge 3$, extending a result by Loray,
Pereira, and Touzet for degree three foliations on $\mathbb P^3$. We show that
the space of codimension one foliations of degree three on $\mathbb{P}^n$,
$n\ge 3$, has exactly $18$ distinct irreducible components parameterizing
foliations without rational first integrals, and at least $6$ distinct
irreducible components parameterizing foliations with rational first integrals.
