  In this paper, we characterize and study dynamical properties of cubic vector
fields on the sphere $\mathbb{S}^2 = \{(x, y, z) \in \mathbb{R}^3 ~|~
x^2+y^2+z^2 = 1\}$. We start by classifying all degree three polynomial vector
fields on $\mathbb{S}^2$ and determine which of them form Kolmogorov systems.
Then, we show that there exist completely integrable cubic vector fields on
$\mathbb{S}^2$ and also study the maximum number of various types of invariant
circles for homogeneous cubic vector fields on $\mathbb{S}^2$. We find a tight
bound in each case. Further, we also discuss phase portraits of certain cubic
Kolmogorov vector fields on $\mathbb{S}^2$.
