  This survey explores the geometric perspective on policy optimization within
the realm of feedback control systems, emphasizing the intrinsic relationship
between control design and optimization. By adopting a geometric viewpoint, we
aim to provide a nuanced understanding of how various ``complete
parameterization'' -- referring to the policy parameters together with its
Riemannian geometry -- of control design problems, influence stability and
performance of local search algorithms. The paper is structured to address key
themes such as policy parameterization, the topology and geometry of
stabilizing policies, and their implications for various (non-convex) dynamic
performance measures. We focus on a few iconic control design problems,
including the Linear Quadratic Regulator (LQR), Linear Quadratic Gaussian (LQG)
control, and $\mathcal{H}_\infty$ control. In particular, we first discuss the
topology and Riemannian geometry of stabilizing policies, distinguishing
between their static and dynamic realizations. Expanding on this geometric
perspective, we then explore structural properties of the aforementioned
performance measures and their interplay with the geometry of stabilizing
policies in presence of policy constraints; along the way, we address issues
such as spurious stationary points, symmetries of dynamic feedback policies,
and (non-)smoothness of the corresponding performance measures. We conclude the
survey with algorithmic implications of policy optimization in feedback design.
