Flow problems with moving boundaries and interfaces include fluid--structure interaction (FSI) and a number of other classes of problems, have an important place in engineering analysis and design, and offer some formidable computational challenges. Bringing solution and analysis to such flow problems motivated the development of the Deforming-Spatial-Domain/Stabilized Space--Time (DSD/SST) method. Since its inception, the DSD/SST method and its improved versions have been applied to a diverse set of challenging problems with a common core computational technology need. The classes of problems solved include free-surface and two-fluid flows, fluid--object and fluid--particle interaction, FSI, and flows with solid surfaces in fast, linear or rotational relative motion. Some of the most challenging FSI problems, including parachute FSI and arterial FSI, are being solved and analyzed with the DSD/SST method as a core technology. Better accuracy and improved turbulence modeling were brought with the recently-introduced variational multiscale (VMS) version of the DSD/SST method, which is called DSD/SST-VMST (also ST-VMS). In specific classes of problems, such as parachute FSI, arterial FSI, aerodynamics of flapping wings, and wind-turbine aerodynamics, the scope and accuracy of the FSI modeling were increased with the special ST FSI techniques targeting each of those classes of problems. This article provides an overview of the core ST FSI technique, its recent versions, and the special ST FSI techniques. It also provides examples of challenging problems solved and analyzed in parachute FSI, arterial FSI, aerodynamics of flapping wings, and wind-turbine aerodynamics.