A symmetric form is a multivariate polynomial that is symmetric in the variables and its monomials have the same degree. Symmetric forms are ubiquitous in various areas of mathematics, including combinatorics, invariant theory, inequalities and so on. In the study and the application of symmetric forms, one of the crucial challenge is to understand the structure of their critical points over the unit sphere. In this talk, we report on a recent progress on this challenge. We begin by showing how to partition the set of critical points into ``cells'' so that every critical point in a cell has the same combinatorial structure. Then, by applying the obtained partition, we tackled the following tasks: \begin{itemize} \item Determine the number of critical points up to symmetry. \item Determine the structure of an optimal critical point. \end{itemize} For symmetric forms of degree 1 and 2, the tasks are trivial. However, for symmetric forms of degree 3, the tasks are non-trivial. Hence, we tackled the smallest non-trivial case (degree 3) and report on the findings. First, we briefly provide some notations in order to state our main theorems. Note that the critical set $C$ has the following obvious symmetries. \begin{itemize} %\item $c\in C\ \ \Longleftrightarrow\ \ -c\in C$ \item $c\in C\ \ \Longleftrightarrow\ \ \pi\left( c\right) \in C\ \ \ \ \ \ $for every permutation $\pi$. \end{itemize} \noindent We write $c\equiv c^{\prime}$ if $c$ and $c^{\prime}$ are equivalent up to the symmetry, that is $\,c^{\prime}% =\pi\left( c\right) $ for some $\pi$. Thus let $\#\overline{C}$ denote the number of the critical points up to symmetry. \medskip The main results are the following two theorems. \begin{theorem} [Combinatorics]Let $f\in\mathbb{R}\left[ x_{1},\ldots,x_{n}\right] $ be a symmetric form of degree $d=3$. If the critical set $C$ is finite, then we have% \[ \#\overline{C}\ =1+\#\ \text{of positives in }\left( D_{1,n-1},\ldots ,D_{n-1,1}\right), \] where% \[ D_{pq}=9a_{3}^{2}-4pq\left( a_{21}^{2}+3a_{111}\left( 3a_{3}+na_{21}\right) \right). \] \end{theorem} \medskip \noindent The second main theorem is derived from the following research history. In 1987, Choi, Lam and Reznick made a systematic study of even symmetric sextics. Their result implies that the optimal value of symmetric cubic forms on the standard simplex is always obtained at the center of some facets,% \[ (0,\cdots,0,\underset{q}{\underbrace{\frac{1}{q},\cdots,\frac{1}{q}}% }),\ \ \ 1\leq q\leq n. \] What would happen if the standard simplex were replaced by the unit sphere? The answer is on $(s,t,\cdots,t)$. \begin{theorem}[Optimal] Let $f\in\mathbb{R}\left[ x_{1},\ldots,x_{n}\right] $ be a symmetric form of degree $d=3$. There is a maximizer (minimizer) of $f$ over the unit sphere $\mathbb{S}% ^{n-1}$ with type $(s,t,\cdots,t)$ ($s=t$ $s=t$ is allowed). \end{theorem}