Closed orbits on symmetric spheres and dynamical convexity

I will survey some results on the multiplicity and stability of closed orbits of Reeb flows on the standard contact sphere. After that, I will discuss recent results, obtained in a joint work with V. Ginzburg, on the multiplicity of closed orbits on symmetric strongly dynamically convex spheres and examples of dynamically convex contact forms that are not equivalent to convex ones via a contactomorphism commuting with the antipodal map.