Typical unsteady vortex-dominated flows like those involved in bio-inspired propulsion, airfoil separation, bluff body wakes, and vortex-induced vibrations can be prohibitively expensive to simulate and impossible to measure comprehensively. These examples are governed by nonlinear interactions, and often involve moving boundaries, high-dimensional parameter spaces, and multiscale flow structures. The classical way to get around these challenges has been to reduce the experimental complexity by using canonical motions or simplified unsteady inflow conditions. A paradigm shift is emerging in the form of self-exploring automated experiments that combine the automation of the experimental pipeline with data-science tools to increase experimental throughput and expedite scientific discovery. Such automated experiments can explore and exploit higher-dimensional parameter spaces and cover more realistic and technically relevant unsteady conditions compared to what is traditionally feasible with supervised canonical experiments. This alternative approach can yield robust and generalizable models and control solutions, as well as the discovery of rare and extreme events. Here, we provide a perspective on the transformative potential of self-exploring automated experiments for the discovery, optimization, and control of unsteady vortex-dominated flow phenomena.