We will showcase persistent homology as a promising new tool for use in the study of complicated fluid flows. Through a collection of examples spanning 2D Kolmogorov and Rayleigh-Bénard convection flows to fully-developed 3D turbulence and combustion dynamics, we will demonstrate how persistent homology can be useful for: symmetry and dimension reduction; pattern identification and analysis; and large-scale data organization. These examples include both established results and new directions, and present the cumulative joint work of many researchers who are either currently affiliated with or have ongoing collaborations with professors at Rutgers University (Konstantin Mischaikow), Virginia Tech (Mark Paul), or Georgia Tech (Michael Schatz).