Quadratic optimization is a major step toward global optimization. The indefiniteness of a quadratic form may lead to NP-Hardness. Variants of quadratic optimization are often seen in applications such as the linearly constrained quadratic programming problem, trust region method, max-cut problem, binary quadratic programming problem, box constrained quadratic programming problem, and quadratically constrained quadratic programming problem. In this talk, we report the research progress on quadratic optimization made by integrating the Lagrangian-based methods, semi-definite programming techniques, conic programming theory, and cones of nonnegative quadratic functions.