In some sense, linear-quadratic optimal control problem is well-understood: Under some convex conditions, optimal control uniquely exists and can be represented as a linear state feedback via the solution to a Riccati equation. A key assumption is the weighting matrix of the control in the cost functional is (uniformly) positive definite. A natural question is whether such a condition necessary? Some relevant questions are: What is the conceptual difference between open-loop and closed-loop (optimal) controls? What are the relationships among optimal open-loop, closed-loop optimal controls, and the corresponding two-point boundary value problems, the Riccati equation, etc. The similar questions can be asked for stochastic LQ problems, and for two-person zero-sum LQ differential games. In this talk, we will present some of our recent works along this line. This talk is based on the joint works of mine with Jingrui Sun.