At the start of the millennium a breakthrough in the overlap of semi-algebraic geometry, sum-of-squares (SOS) polynomial optimization and computing has provided a constructive method for solving a plethora of problems for systems whose dynamics can be described by polynomial functions.The talk will give a broad introduction to SOS optimization and to the general ideas of its applications in fluid dynamics. This will be followed by examples of applications, including examples on the global stability of fluid flows, bounds for long-time averages of deterministic and stochastic systems, control, input-output analysis, and determination of periodic orbits. The talk will be concluded with a more detailed overview of the ongoing research on application of the same ideas to accelerating time averaging in numerical calculations.